MathDB

In a triangle

ABC

, it is drawn a circumference with center in the incenter

I

and that meet twice each of the sides of the triangle: the segment

BC

D

and

P

(where

D

is nearer two

B

); the segment

CA

E

and

Q

(where

E

is nearer to

C

); and the segment

AB

F

and

R

( where

F

is nearer to

A

).
Let

S

be the point of intersection of the diagonals of the quadrilateral

EQFR

. Let

T

be the point of intersection of the diagonals of the quadrilateral

FRDP

. Let

U

be the point of intersection of the diagonals of the quadrilateral

DPEQ

.
Show that the circumcircle to the triangle

\triangle{FRT}

\triangle{DPU}

and

\triangle{EQS}

have a unique point in common.

Problem Statement