Question 183
Let $\mathcal{K}$ be a circle in the plane, and $\mathcal{K}_1$ and $\mathcal{K}_2$ be disjoint circles that touch the circle $\mathcal{K}$ from the inside at points $A$ and $B$. Let $t$ be the common tangent of the circles $\mathcal{K}_1$ and $\mathcal{K}_2$, which touches them at the points $C$ and $D$ such that $\mathcal{K}_1$ and $\mathcal{K}_2$ are both on the same bank of the line $t$, and the center of the circle $\mathcal{K}$ on its other bank. We denote by $E$ the intersection of the lines $A C$ and $B D$. Prove that the point $E$ lies on the circle $\mathcal{K}$.