KöMaL 2013 B4551

$P$ is the point on base $AB$ of a symmetric trapezium ABCD for which $AP$-$BP$=$AC$-$BC$. The perpendicular drawn to $AB$ at $P$ intersects the lines $CD$, $AC$ and $BD$ at the points $Q$, $R$ and $S$, respectively. Let $k_1$ be the circle that touches the lines $AC$ and $BD$ at points $R$ and $S$, and let $k_2$ be the circle of diameter $PQ$. Prove that the circles $k_1$ and $k_2$ touch each other.