Let $C$ and $D$ be different points on the semicircle with diameter $AB$. The lines $AC$ and $BD$ intersect at $E$, and the lines $AD$ and $BC$ intersect at $F$. Prove that the midpoints $X,Y,Z$ of the segments $AB,CD,EF$ respectively are collinear.
Source: Slovenia 1997 2nd Grade P3
Tags: geometry
Let $C$ and $D$ be different points on the semicircle with diameter $AB$. The lines $AC$ and $BD$ intersect at $E$, and the lines $AD$ and $BC$ intersect at $F$. Prove that the midpoints $X,Y,Z$ of the segments $AB,CD,EF$ respectively are collinear.