IR-TST 2011 Problem 6

The circle $\omega$ with center $O$ has given. From an arbitrary point $T$ outside of $\omega$ draw tangents $TB$ and $TC$ to it. $K$ and $H$ are on $TB$ and $TC$ respectively.a) $B'$ and $C'$ are the second intersection point of $OB$ and $OC$ with $\omega$ respectively. $K'$ and $H'$ are on angle bisectors of $\angle BCO$ and $\angle CBO$ respectively such that $KK' \bot BC$ and $HH'\bot BC$ . Prove that $K,H',B'$ are collinear if and only if $H,K',C'$ are collinear.b) Consider there exist two circle in $TBC$ such that they are tangent two each other at $J$ and both of them are tangent to $\omega$ .and one of them is tangent to $TB$ at $K$ and other one is tangent to $TC$ at $H$ . Prove that two quadrilateral $BKJI$ and $CHJI$ are cyclic ( $I$ is incenter of triangle $OBC$ ).