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equal segments wanted, circles passing through incenter related

Source: 2019 Austrian Federal Competition For Advanced Students, Part 1 p2

March 4, 2020

geometryincentercirclesequal segments

Problem Statement

Let

ABC

be a triangle and

I

its incenter. The circle passing through

A, C

and

I

intersect the line

BC

for second time at point

X

. The circle passing through

B, C

and

I

intersects the line

AC

for second time at point

Y

. Show that the segments

AY

and

BX

have equal length.