2022 SSMO Speed Round Problems/Problem 4

Problem

Consider a quadrilateral $ABCD$ with area $120$ and satisfying $AB+CD=AD+BC=24$. There exists a point $P$ in 3D space such that the distances from $P$ to $AB$, $BC$, $CD$, and $DA$ are all equal to $13$. Find the volume of $PABCD$.

Solution