Let
![T_1](/media/m/4/0/c/40c71cac72788d0bef7cc411fb894e9e.png)
be a triangle having
![a, b, c](/media/m/9/e/9/9e9dfe78930065fbe5a777e9b07c27c4.png)
as lengths of its sides and let
![T_2](/media/m/e/3/5/e35542d52d98e84ff8224ac52cb086e2.png)
be another triangle having
![u, v,w](/media/m/4/f/f/4ff9583fbd4bc007903711389918f657.png)
as lengths of its sides. If
![P,Q](/media/m/4/7/3/473c1fde1bc2a906fbfdc57f588be4d5.png)
are the areas of the two triangles, prove that
![16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).](/media/m/8/1/d/81dd572e369daeca618e65ced847eb59.png)
When does equality hold?
%V0
Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that
$$16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).$$
When does equality hold?