KöMaL 2015 B4730

The circles $k_1$ and $k_2$ touch at point $E$ . Points $X_i$ and $Y_i$ are marked on each circle $k_i$ ($i = 1,2$) such that the two lines $X_iY_i$ intersect each other on the common interior tangent of the circles. Prove that the line connecting the centres of circles $X_1X_2E$ and $Y_1Y_2E$ , and the other line connecting the centres of circles $X_1Y_2E$ and $X_2Y_1E$ also intersect each other on the common interior tangent of the circles.