Question 57
Let $P$ be a point in the interior of the acute triangle $ABC$ and let $D$, $E$, $F$ be respectively the intercepts of lines $AP$, $BP$, $CP$ with the sides $BC$, $CA$, $AB$.
a) Prove that the area of the triangle $DEF$ is not greater than a quarter of the area of triangle $ABC$.
b) Prove that the inradius of $DEF$ is not greater than a quarter of the inradius of triangle $ABC$.
a) Prove that the area of the triangle $DEF$ is not greater than a quarter of the area of triangle $ABC$.
b) Prove that the inradius of $DEF$ is not greater than a quarter of the inradius of triangle $ABC$.