Triangles with Cotangents

Geometry Level 3

In A B C \triangle ABC , let D D be the foot of the altitude from A A down to B C BC . Given that A D 2 ( cot B + cot C ) = 800 AD^{2} (\cot B + \cot C) = 800 , find the area of A B C \triangle ABC .


The answer is 400.

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1 solution

Alan Yan
Sep 22, 2015

A D cot B = B D AD\cot B = BD A D cot C = D C AD\cot C = DC A D 2 ( cot B + cot C = A D ( B D + D C ) = B C A D = 2 K = 800 K = 400 AD^2(\cot B + \cot C = AD(BD + DC) = BC \cdot AD = 2K = 800 \implies K = \boxed{400}

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