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AB + AC >= BC cos A + 2h sin A

Source: Ukraine TST 2010 p7

May 5, 2020
geometrygeometric inequalitytrigonometryaltitude

Problem Statement

Denote in the triangle ABCABC by hh the length of the height drawn from vertex AA, and by α=∠BAC\alpha = \angle BAC. Prove that the inequality AB+AC≥BC⋅cos⁡α+2h⋅sin⁡αAB + AC \ge BC \cdot \cos \alpha + 2h \cdot \sin \alpha . Are there triangles for which this inequality turns into equality?
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