MathNet 2012 Question 66

Let $P$ be a point inside triangle $ABC$. Construct the points $P_1$, $P_2$, $P_3$ such that $PP_1 \perp BC$, $PP_2 \perp CA$, $PP_3 \perp AB$ and $BP_3 = BP_1$, $CP_2 = CP_1$. Prove that $AP_3 = AP_2$.