Školjka

%V0 Let $ABC$ be a triangle, and $P$ a point in the interior of this triangle. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Assume that $AP^{2}+PD^{2}=BP^{2}+PE^{2}=CP^{2}+PF^{2}$. Furthermore, let $I_{a}$, $I_{b}$, $I_{c}$ be the excenters of triangle $ABC$. Show that the point $P$ is the circumcenter of triangle $I_{a}I_{b}I_{c}$.