PL-OM 2002 Problem 5

There is given a triangle $ABC$ in a space. A sphere does not intersect the plane of $ABC$ . There are $4$ points $K, L, M, P$ on the sphere such that $AK, BL, CM$ are tangent to the sphere and $\frac{AK}{AP} = \frac{BL}{BP} = \frac{CM}{CP}$ . Show that the sphere touches the circumsphere of $ABCP$ .