Octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle $\Omega$ with center $O$. It is known that $A_1A_2\|A_5A_6$, $A_3A_4\|A_7A_8$ and $A_2A_3\|A_5A_8$. The circle $\omega_{12}$ passes through $A_1$, $A_2$ and touches $A_1A_6$; circle $\omega_{34}$ passes through $A_3$, $A_4$ and touches $A_3A_8$; the circle $\omega_{56}$ passes through $A_5$, $A_6$ and touches $A_5A_2$; the circle $\omega_{78}$ passes through $A_7$, $A_8$ and touches $A_7A_4$. The common external tangent to $\omega_{12}$ and $\omega_{34}$ cross the line passing through ${A_1A_6}\cap{A_3A_8}$ and ${A_5A_2}\cap{A_7A_4}$ at the point $X$. Prove that one of the common tangents to $\omega_{56}$ and $\omega_{78}$ passes through $X$.

Let$ AB$ and $AC$ be the tangents from a point $A$ to a circle $ \Omega$. Let $M$ be the midpoint of $BC$ and $P$ be an arbitrary point on this segment. A line $AP$ meets $ \Omega$ at points $D$ and $E$. Prove that the common external tangents to circles $MDP$ and $MPE$ meet on the midline of triangle $ABC$.

Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.

Let $\omega_1$ and $\omega_2$ be two circles that intersect at point $A$ and $B$. Define point $X$ on $\omega_1$ and point $Y$ on $\omega_2$ such that the line $XY$ is tangent to both circles and is closer to $B$. Define points $C$ and $D$ the reflection of $B$ WRT $X$ and $Y$ respectively. Prove that the angle $\angle{CAD}$ is less than $90^{\circ}$

Two unequal circles intersect at $X$ and $Y$. Their common tangents intersect at $Z$. One of the tangents touches the circles at $P$ and $Q$. Show that $ZX$ is tangent to the circumcircle of $PXQ$.