Let P a point inside a trapezoid $ABCD$, find $AP \cdot PC$

Question

Let $ABCD$ be an isosceles trapezoid with bases $AB=32$ and $CD=18$. Let $P$ be a point inside $ABCD$ such that $\angle PAD=\angle PBA$ and $\angle PDA=\angle PCD$. Moreover the area of $ABP$ is $192$. Find the product of the lenghts of the segments $PA$ and $PC$.

Answer 1

Since $\angle PAD=\angle PBA$ the line $AD$ is tangent on circumcircle for $ABP$. Since $\angle PDA=\angle PCD$ the line $AD$ is also tangent on circumcircle for $PCD$. So $P$ is one of the intersection point of mentioned circles. Therefore it is easy to construct $P$ and it is not uniquely determined with base length and it is easy to see that $PA\cdot PC$ is not numerical determined (drawings in Geogebra).