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bisector equal to altitude

Source: 2001 Moldova MO Grade 12 P4

April 13, 2021
geometryTriangles

Problem Statement

In a triangle ABCABC, BC=aBC=a, AC=bAC=b, ∠B=β\angle B=\beta and ∠C=γ\angle C=\gamma. Prove that the bisector of the angle at AA is equal to the altitude from BB if and only if b=acos⁡β−γ2b=a\cos\frac{\beta-\gamma}2.
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