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NMO Selection Tests for the Junior Balkan Mathematical Olympiad

Geometry Difficulty 4.8 AIME Prove it Romania

The measure of the angle $\hat{A}$ of the acute triangle $ABC$ is $60^\circ$ , and $HI = HB$ , where $I$ and $H$ are the incenter and the orthocenter of the triangle $ABC$ . Find the measure of the angle $\hat{B}$ .

Solution

We have $m(\angle BIC) \equiv m(\angle BHC) = 120^\circ$ , hence $B$ , $H$ , $I$ , $C$ are situated on a circle.

If $m(\angle B) > 60^\circ$ then $m(\angle HBI) = m(\angle ABI) - m(\angle ABH) = \frac{1}{2}m(\angle B) - 30^\circ$ .

But $m(\angle HBI) = m(\angle HIB) = m(\angle HCB) = 90^\circ - m(\angle B)$ , hence $m(\angle B) =$

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