Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.7 AIME, harder Prove it

B2. Let $A B C$ be an acute-angled triangle and $D$ a point inside this triangle such that $\angle B A D = \angle D C B$ and $\angle C B D = \angle D A C$ . Prove that the lines $A D$ and $B C$ are perpendicular.

Solution

B2. In triangle $ABC$ , the angle $\angle BAC$ is $30^{\circ}$ , and $P$ is a point inside this triangle. The reflections of point $P$ across the sides $BC$ , $CA$ , and $AB$ are denoted as $P_{A}$ , $P_{B}$ , and $P_{C}$ , respectively. Suppose that $P_{A} P_{B} P_{C}$ is an equilateral triangle. Prove that $\angle BPC = 90^{\circ}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.