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Intersection of a Circumcircle and an Incircle

Source:

August 8, 2024

geometrycircumcircle2022

Problem Statement

In equilateral

\triangle ABC

, point

D

lies on

\overline{BC}

such that the radius of the circumcircle

\Gamma_1

of

\triangle ACD

is

7

and the radius of the incircle

\Gamma_2

of

\triangle{ABD}

is

2

. Suppose that

\Gamma_1

and

\Gamma_2

intersect at points

X

and

Y

. Find

XY

.

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