Christmas Streak 79/88: Geometry Rush #Boss

In a triangle $\rm ABC$ with $\angle \rm A = 70^{\circ},$ pick a point $\rm P$ in the interior and $\rm Q$ in the exterior so that $\rm \triangle AQB$ and $\rm \triangle APC$ are similar.

Given that $\rm \overline{AP}\times\overline{BC}=\overline{BP}\times\overline{CA}=\overline{CP}\times\overline{AB},$ find the size of $\angle \rm BPC$ in degrees.

This problem is a part of <Christmas Streak 2017> series .