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Concurrent perpendicular bisectors from three circles in a triangle

Source: German TST 2022, probably on exam 6-7, again proposed by me

July 19, 2022

geometryperpendicular bisectorTriangle GeometryTST

Problem Statement

Given a triangle

ABC

and three circles

x

,

y

and

z

such that

A \in y \cap z

,

B \in z \cap x

and

C \in x \cap y

.
The circle

x

intersects the line

AC

at the points

X_b

and

C

, and intersects the line

AB

at the points

X_c

and

B

. The circle

y

intersects the line

BA

at the points

Y_c

and

A

, and intersects the line

BC

at the points

Y_a

and

C

. The circle

z

intersects the line

CB

at the points

Z_a

and

B

, and intersects the line

CA

at the points

Z_b

and

A

. (Yes, these definitions have the symmetries you would expect.)
Prove that the perpendicular bisectors of the segments

Y_a Z_a

,

Z_b X_b

and

X_c Y_c

concur.

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