Given a trapezoid $ABCD$, with $AB\parallel CD$. Lines $AC$ and $BD$ intersect at point $E$, and lines $AD$ and $BC$ intersect at point $F$. It turns out that the circle with diameter $EF$ is tangent to the midline of the trapezoid. Prove that there exists a square such that there is a mutual correspondence between all six lines containing pairs of its vertices, and points $A$, $B$, $C$, $D$, $E$, and $F$: each line corresponds to a point lying on it. Proposed by V. Mokin