Question 122
$M$ is an arbitrary point on the side $BC$ of the triangle $ABC$. $\omega$ is a circle tangent to the segments $AB$ and $BM$ in $T$ and $K$ respectively, also is tangent to the circumcircle of $AMC$ in $P$. If $TK \parallel AM$, prove that the circumcircles of $APT$ and $KPC$ are tangent to each other.