MathDB

Convex circumscribed quadrilateral

ABCD

with its incenter

I

is given such that its incircle is tangent to

\overline{AD},\overline{DC},\overline{CB},

and

\overline{BA}

K,L,M,

and

N

. Lines

\overline{AD}

and

\overline{BC}

meet at

E

and lines

\overline{AB}

and

\overline{CD}

meet at

F

. Let

\overline{KM}

intersects

\overline{AB}

and

\overline{CD}

X,Y

, respectively. Let

\overline{LN}

intersects

\overline{AD}

and

\overline{BC}

Z,T

, respectively. Prove that the circumcircle of triangle

\triangle XFY

and the circle with diameter

EI

are tangent if and only if the circumcircle of triangle

\triangle TEZ

and the circle with diameter

FI

are tangent. Proposed by Mahdi Etesamifard

Problem Statement