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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+ | == The Actual Problem == | ||
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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 HQA = 90◦</math> and let <math>K</math> be the point on <math>\gamma</math> such that <math>\angle HKQ = 90◦</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. | ||
==Solution== | ==Solution== |
Revision as of 13:58, 1 June 2024
Let be an acute triangle with
. Let
be its circumcircle,
its orthocenter, and
the foot of the altitude from
. Let
be the midpoint of
. Let
be the point on
such that
. Assume that the points
,
,
,
, and
are all different, and lie on
in this order.
Prove that the circumcircles of triangles and
are tangent to each other.
The Actual Problem
Let be an acute triangle with
. Let
be its circumcircle,
its orthocenter, and
the foot of the altitude from
. Let
be the midpoint of
. Let
be the point on
such that
and let
be the point on
such that
. Assume that the points
,
,
,
and
are all different and lie on
in this order. Prove that the circumcircles of triangles
and
are tangent to each other.
Solution
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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 |