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Difference between revisions of "2015 IMO Problems/Problem 3"

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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.
 
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.
  
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==See Also==
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==Solution==
  
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{{IMO box|year=2015|num-b=2|num-a=4}}
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[[File:IMO2015 P3.png|600px|up]]
  
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[[Category:Olympiad Geometry Problems]]
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We know that there is a negative inversion which is at <math>H</math> and swaps the nine-point circle and <math>\Gamma</math>. And this maps:
  
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Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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<math>A \longleftrightarrow F</math>. Also, let <math>M \longleftrightarrow Q`</math>. Of course <math>\triangle HFM \sim \triangle HQ'A</math> so <math>\angle HQ'A = 90</math>. Hence, <math>Q' = Q</math>. So:
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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<math>M \longleftrightarrow Q</math>. Let <math>HA</math> and <math>HQ</math> intersect with nine-point circle <math>T</math> and <math>Q</math>, respectively. Let's define the point <math>L</math> such that <math>TNML</math> is rectangle. We have found <math>M \longleftrightarrow Q</math> and if we do the same thing, we find:
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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<math>L \longleftrightarrow K</math>. Now, we can say:
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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<math>(KQH) \longleftrightarrow ML</math> and <math>(FKM) \longleftrightarrow (ALQ)</math>. İf we manage to show <math>ML</math> and <math>(ALQ)</math> are tangent, the proof ends.  
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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We can easily say <math>TN || AQ</math> and <math>AQ = 2.TN</math> because <math>T</math> and <math>N</math> are the midpoints of <math>HA</math> and <math>HQ</math>, respectively.  
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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Because of the rectangle <math>TNML</math>, <math>TN || ML</math> and <math>TN = ML</math>.  
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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Hence, <math>ML || AQ</math> and <math>AQ = 2.ML</math> so <math>L</math> is on the perpendecular bisector of <math>AQ</math> and that follows <math>\triangle ALQ</math> is isoceles. And we know that <math>ML || AQ</math>, so <math>ML</math> is tangent to <math>(ALQ)</math>. We are done. <math>\blacksquare</math>
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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~ EgeSaribas
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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Really Important Note: This solution is in the "IMO 2015 Solution Notes" which is written by Evan Chen.  
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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There is the link: https://web.evanchen.cc/exams/IMO-2015-notes.pdf
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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{{solution}}
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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==See Also==
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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{{IMO box|year=2015|num-b=2|num-a=4}}
  
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
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[[Category:Olympiad Geometry Problems]]
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.
 

Latest revision as of 15:02, 1 June 2024

Let $ABC$ be an acute triangle with $AB>AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HKQ=90^\circ$. Assume that the points $A$, $B$, $C$, $K$, and $Q$ are all different, and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Solution

up

We know that there is a negative inversion which is at $H$ and swaps the nine-point circle and $\Gamma$. And this maps:

$A \longleftrightarrow F$. Also, let $M \longleftrightarrow Q`$. Of course $\triangle HFM \sim \triangle HQ'A$ so $\angle HQ'A = 90$. Hence, $Q' = Q$. So:

$M \longleftrightarrow Q$. Let $HA$ and $HQ$ intersect with nine-point circle $T$ and $Q$, respectively. Let's define the point $L$ such that $TNML$ is rectangle. We have found $M \longleftrightarrow Q$ and if we do the same thing, we find:

$L \longleftrightarrow K$. Now, we can say:

$(KQH) \longleftrightarrow ML$ and $(FKM) \longleftrightarrow (ALQ)$. İf we manage to show $ML$ and $(ALQ)$ are tangent, the proof ends.

We can easily say $TN || AQ$ and $AQ = 2.TN$ because $T$ and $N$ are the midpoints of $HA$ and $HQ$, respectively.

Because of the rectangle $TNML$, $TN || ML$ and $TN = ML$.

Hence, $ML || AQ$ and $AQ = 2.ML$ so $L$ is on the perpendecular bisector of $AQ$ and that follows $\triangle ALQ$ is isoceles. And we know that $ML || AQ$, so $ML$ is tangent to $(ALQ)$. We are done. $\blacksquare$

~ EgeSaribas

Really Important Note: This solution is in the "IMO 2015 Solution Notes" which is written by Evan Chen.

There is the link: https://web.evanchen.cc/exams/IMO-2015-notes.pdf

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See Also

2015 IMO (Problems) • Resources
Preceded by
Problem 2 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 4
All IMO Problems and Solutions
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