OFM 2023 J3

Let $\Gamma$ and $\Gamma'$ be two circles with centres $O$ and $O'$ , such that $O$ belongs to $\Gamma'$ . Let $M$ be a point on $\Gamma'$ , outside of $\Gamma$ . The tangents to $\Gamma$ that go through $M$ touch $\Gamma$ in two points $A$ and $B$ , and cross $\Gamma'$ again in two points $C$ and $D$ . Finally, let $E$ be the crossing point of the lines $AB$ and $CD$ . Prove that the circumcircles of the triangles $CEO'$ and $DEO'$ are tangent to $\Gamma'$ .