MathDB

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

Problem Statement