KöMaL 2012 A572

Two circles $k_1$ and $k_2$ with centres $O_1$ and $O_2$, respectively, intersect perpendicularly at $P$ and $Q$. Their external homothety center is $H$. The line $t$is tangent to $k_1$ at $T_1$ and tangent to $k_2$ at $T_2$. Let $X$ be a point in the interior of the two circles such that $HX$=$HP$=$HQ$, and let $X$' be the reflection of $X$ about $t$. Let the circle $XX$'$T_2$ and the shorter arc $PQ$ of $k_1$ meet at $U_1$, and let the circle $XX$'$T_1$ and the shorter arc $PQ$ of $k_2$ meet at $U_2$. Finally, let the lines $O_1 U_1$ and $O_2 U_2$ meet at $V$. Show that $VU_1$=$VU_2$.