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Estonian Math Competitions

Geometry Difficulty 4.4 AIME Prove it Estonia

Let $ABC$ be a triangle where $|AB| = |AC|$ . Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$ , respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$ .

Solution

Without loss of generality, we can assume that $|AP| \le |AQ|$ . Let $O$ be the circumcenter of $ABC$ . Let $R$ be the intersection point of the bisector of $\angle BAC$ with the circumcircle of the triangle $PAQ$ — we then have $|RB| = |RC|$ (Fig. 21).

Also, $\angle APR = 180^\circ - \angle AQR = \angle CQR$ and $|RP| = |RQ|$ (since $\angle RAP = \angle RAQ$ ). So, $|AP| = |CQ| \iff \triangle APR \cong \triangle CQR \iff |RA| = |RC| \iff R = O$ (where $|RA| = |RC| \Rightarrow \triangle APR \cong \triangle CQR$ by two sides and obtuse angle).

Fig. 21

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