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MS = MT iff X,Y,Z,W are concyclic, starting with 2 intersecting circles

Source: 2011 Brasil IMO TST 4.2

July 23, 2021

Concyclicgeometryequal segments

Problem Statement

Given two circles

\omega_1

and

\omega_2

, with centers

\omega_1

and

XA

, respectively intesrecting at two points

A

and

B

. Let

X

and

Y

be points on

\omega_1

. The lines

XA

and

YA

intersect

\omega_2

again in

Z

and

W

, respectively, such that

A

is between

X,Z

and

A

is between

Y,W

. Let

M

be the midpoint of

O_1O_2

, S be the midpoint of

XA

and

T

be the midpoint of

WA

. Prove that

MS = MT

if, and only if, the points

X, Y, Z

and

W

are concyclic.

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