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Slovenian Mathematical Olympiad, 2008

Geometry Difficulty 4.5 AIME Prove it Slovenia

Let $D$ be a point on the side $AB$ of an acute triangle $ABC$ such that the triangle $BCD$ is also acute. Denote the orthocentre of the triangle $BCD$ by $H$ . Prove: if points $A$ , $D$ , $H$ and $C$ are concyclic, then the triangle $ABC$ is isosceles.

Solution

Let $\angle BAC = \alpha$ . Since $A$ , $D$ , $H$ and $C$ are concyclic, we have $\angle DHC = \pi - \angle BAC = \pi - \alpha$ .

Let $E$ be the foot of the altitude from $C$ to the side $BD$ . Since $\angle DHE = \pi - \angle DHC = \alpha$ , we have $\angle HDB = \frac{\pi}{2} - \alpha$ .

Since the line $DH$ is perpendicular to the side $BC$ , we have $\angle CBA = \frac{\pi}{2} - \angle BDH = \alpha$ , so the triangle $ABC$ is isosceles.

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