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Prove that MS=MT iff X,Y,Z,W are concylic

Source: Saudi Arabia IMO TST Day III Problem 4

July 22, 2014

geometry unsolvedgeometry

Problem Statement

Let

\omega_1

and

\omega_2

with center

O_1

and

O_2

respectively, meet at points

A

and

B

. Let

X

and

Y

be points on

\omega_1

. Lines

XA

and

Y A

meet

\omega_2

at

Z

and

W

, respectively, such that

A

lies between

X

and

Z

and between

Y

and

W

. Let

M

be the midpoint of

O_1O_2

,

S

be the midpoint of

XA

and

T

be the midpoint of

W A

. Prove that

MS = MT

if and only if

X,~ Y ,~ Z

and

W

are concyclic.

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