MathNet 2011 Question 65

Let $ABC$ be a triangle such that $|AB| = |AC|$. Let $D$ be a point on the side $AC$ such that $|AD| \lt |CD|$, and let $P$ be a point on the segment $BD$ such that $\angle APC = 90^\circ$. If $\angle ABP = \angle BCP$, determine $|AD| : |CD|$.
(Stipe Vidak)