MathDB

Let

ABC

be an acute triangle with

AB<AC<BC

and let

D

be a point on it's extension of

BC

towards

C

. Circle

c_1

, with center

A

and radius

AD

, intersects lines

AC,AB

and

CB

at points

E,F

, and

G

respectively. Circumscribed circle

c_2

of triangle

AFG

intersects again lines

FE,BC,GE

and

DF

at points

J,H,H'

and

J'

respectively. Circumscribed circle

c_3

of triangle

ADE

intersects again lines

FE,BC,GE

and

DF

at points

I,K,K'

and

I'

respectively. Prove that the quadrilaterals

HIJK

and

H'I'J'K '

are cyclic and the centers of their circumscribed circles coincide.
by Evangelos Psychas, Greece

G5