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Geometry Difficulty 4.6 AIME Prove it India

In a triangle $ABC$ with $B = 90^\circ$ , $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\angle ADB = \varphi$ then prove that $\tan^2(\varphi/2) = \tan(C/2)$ . $\square$

Solution

This follows easily from formulae for inradius of a triangle and simple trigonometric identities. $\square$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.