parmenides51 wrote: Let $P$ be a point in the interior of a triangle $ABC$ and let $d_a,d_b,d_c$ be its distances to $BC,CA,AB$ respectively. Prove that max $(AP, BP, CP) \ge d_a^2+d_b^2+d_c^2$ This cannot be right, as it is not homogeneous... if we scale everything with a factor $\lambda $ then the LHS is of order $\lambda$ and the RHS of order $\lambda^2$, so $\lambda \to \infty$ leads to a contradiction.

oups, I forgot the square root, thanks

Bulgaria TST 2008