Geometry · Triangle geometry · Altitudes · Orthocenter · Cyclic quadrilaterals · Reflection symmetry
Let \(ABC\) be an acute triangle and let \(D\) be the foot of the altitude from \(A\), so that \(D\) lies on the side \(BC\). A point \(P\) is chosen on the segment \(AD\) in such a way that
\[ \angle PBA = \angle PCA . \]
Prove that \(ABC\) is isosceles, or else \(P\) is the orthocenter of \(ABC\).
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Serbian National Competition (Drzavno takmicenje) 2012, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source