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| − | [[Image:Euler Line.PNG|thumb|500px|right|Triangle ''ABC'' with the nine point circle in light orange]] | + | #REDIRECT[[Nine-point circle]] |
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| − | The '''nine point circle''' (also known as ''Euler's circle'' or ''Feuerbach's circle'') of a given [[triangle]] is a circle which passes through 9 "significant" points:
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| − | * The three feet of the [[altitude]]s of the triangle.
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| − | * The three [[midpoint]]s of the [[edge]]s of the triangle.
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| − | * The three midpoints of the segments joining the [[vertex | vertices]] of the triangle to its [[orthocenter]]. (These points are sometimes known as the [[Euler point]]s of the triangle.)
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| − | That such a circle exists is a non-trivial theorem of Euclidean [[geometry]].
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| − | The center of the nine point circle is the [[nine-point center]] and is usually denoted <math>N</math>.
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| − | ==First Proof of Existence==
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| − | Since <math>O_c</math> is the midpoint of <math>AB</math> and <math>E_b</math> is the midpoint of <math>BH</math>, <math>O_cE_b</math> is parallel to <math>AH</math>. Using similar logic, we see that <math>O_bE_c</math> is also parallel to <math>AH</math>. Since <math>E_b</math> is the midpoint of <math>HB</math> and <math>E_c</math> is the midpoint of <math>BC</math>, <math>E_bE_c</math> is parallel to <math>BC</math>, which is perpendicular to <math>AH</math>. Similar logic gives us that <math>O_bO_c</math> is perpendicular to <math>AH</math> as well. Therefore <math>O_bO_cE_bE_c</math> is a rectangle, which is a cyclic figure. The diagonals <math>O_bE_b</math> and <math>O_cE_c</math> are diagonals of the circumcircle. Similar logic to the above gives us that <math>O_aO_cE_aE_c</math> is a rectangle with a common diagonal to <math>O_bO_cE_bE_c</math>. Therefore the circumcircles of the two rectangles are identical. We can also gain that rectangle <math>O_aO_bE_aE_b</math> is also on the circle.
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| − | We now have a circle with the points <math>O_a</math>, <math>O_b</math>, <math>O_c</math>, <math>E_a</math>, <math>E_b</math>, and <math>E_c</math> on it, with diameters <math>O_aE_A</math>, <math>O_bE_b</math>, and <math>O_cE_c</math>. We now note that <math>\angle E_aH_aO_a=\angle E_bH_bO_b=\angle E_cH_cO_c=90^{\circ}</math>. Therefore <math>H_a</math>, <math>H_b</math>, ad <math>H_c</math> are also on the circle. We now have a circle with the midpoints of the sides on it, the three midpoints of the segments joining the vertices of the triangle to its orthocenter on it, and the three feet of the altitudes of the triangle on it. Therefore the nine points are on the circle, and the nine-point circle exists.
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| − | ==Second Proof of Existence==
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| − | We know that the reflection of the orthocenter about the sides and about the midpoints of the triangle's sides lie on the circumcircle. Thus, consider the homothety centered at <math>H</math> with ratio <math>-\frac{1}{2}</math>. It maps the circumcircle of <math>\triangle ABC</math> to the nine point circle, and the vertices of the triangle to its [[Euler point]]s.
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| − | Hence proved.
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| − | {{stub}}
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| − | [[Category:Definition]]
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