Question 27
(GBR 4) The triangle $ABC$ is acute-angled. Let $L$ be any line in the plane of the triangle and let $u,v,w$ be the lengths of the perpendiculars from $A,B,C$ respectively to $L$. Prove that $$u^2\tan A+v^2\tan B+w^2\tan C\ge2\Delta,$$ where $\Delta$ is the area of the triangle, and determine the lines $L$ for which equality holds.