MathDB

Hard geometry

Source: Iran TST 2015 ,exam 1, day 2 problem 3

May 11, 2015

geometrycircumcircle

Problem Statement

ABCD

is a circumscribed and inscribed quadrilateral.

O

is the circumcenter of the quadrilateral.

E,F

and

S

are the intersections of

AB,CD

,

AD,BC

and

AC,BD

respectively.

E'

and

F'

are points on

AD

and

AB

such that

A\hat{E}E'=E'\hat{E}D

and

A\hat{F}F'=F'\hat{F}B

.

X

and

Y

are points on

OE'

and

OF'

such that

\frac{XA}{XD}=\frac{EA}{ED}

and

\frac{YA}{YB}=\frac{FA}{FB}

.

M

is the midpoint of arc

BD

of

(O)

which contains

A

. Prove that the circumcircles of triangles

OXY

and

OAM

are coaxal with the circle with diameter

OS

.

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