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Problem 876

AIME late

Geometry Difficulty 5.7 Prove it

\section*{Problem 2 - 241022}

For a triangle $A B C$ , it is assumed that it is not obtuse and that for the height $C D$ perpendicular to $A B$ , the equation $C D \cdot A C = A D \cdot B C$ holds.

Prove that these assumptions uniquely determine the size $\gamma$ of the interior angle $\angle A C B$ ! Determine this angle size $\gamma$ !

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

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According to the condition, the triangles $A D C$ and $D B C$ are similar, as they match in the ratio of two sides and the angle opposite the larger side. Therefore, $\gamma=\alpha+\beta=90^{\circ}$ .

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