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2014 USAMO Problems/Problem 5: Difference between revisions

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==Problem==
==Problem==
Let <math>ABC</math> be a triangle with orthocenter <math>H</math> and let <math>P</math> be the second intersection of the circumcircle of triangle <math>AHC</math> with the internal bisector of the angle <math>\angle BAC</math>.  Let <math>X</math> be the circumcenter of triangle <math>AHC</math> and <math>Y</math> the orthocenter of triangle <math>APC</math>.  Prove that the length of segment <math>XY</math> is equal to the circumradius of triangle <math>ABC</math>.
Let <math>ABC</math> be a triangle with orthocenter <math>H</math> and let <math>P</math> be the second intersection of the circumcircle of triangle <math>AHC</math> with the internal bisector of the angle <math>\angle BAC</math>.  Let <math>X</math> be the circumcenter of triangle <math>APB</math> and <math>Y</math> the orthocenter of triangle <math>APC</math>.  Prove that the length of segment <math>XY</math> is equal to the circumradius of triangle <math>ABC</math>.


==Solution==
==Solution==

Revision as of 19:51, 6 December 2014

Problem

Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\angle BAC$. Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$. Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.

Solution

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