MathDB

Given trapezoid

ABCD

with parallel sides

AB

and

CD

, assume that there exist points

E

on line

BC

outside segment

BC

, and

F

inside segment

AD

such that \angle DAE \equal{} \angle CBF. Denote by

I

the point of intersection of

CD

and

EF

, and by

J

the point of intersection of

AB

and

EF

. Let

K

be the midpoint of segment

EF

, assume it does not lie on line

AB

. Prove that

I

belongs to the circumcircle of

ABK

if and only if

K

belongs to the circumcircle of

CDJ

. Proposed by Charles Leytem, Luxembourg

Problem Statement