KöMaL 2005 B3832

$P$ is an arbitrary point of the hypotenuse $AB$ of a right-angled triangle ABC . The foot of the altitude drawn from vertex $C$ is $C_1$. The projection of $P$ onto the leg $AC$ is $A_1$, and its projection onto the leg $BC$ is $B_1$. $a$) Prove that the points $P$, $A_1$, $C$, $B_1$, $C_1$ lie on a circle. $b$) Prove that the triangles $A_1 B_1 C_1$ and ABC are similar.