Question 15
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that
$$ AP^{2}+PD^{2}=BP^{2}+PE^{2}=CP^{2}+PF^{2}. $$
Denote by $ I_{A} $, $ I_{B} $, $ I_{C} $ the excentres of the triangle ABC. Prove that P is the circumcentre of the triangle $ I_{A}I_{B}I_{C} $.
$$ AP^{2}+PD^{2}=BP^{2}+PE^{2}=CP^{2}+PF^{2}. $$
Denote by $ I_{A} $, $ I_{B} $, $ I_{C} $ the excentres of the triangle ABC. Prove that P is the circumcentre of the triangle $ I_{A}I_{B}I_{C} $.