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Moving circles touch in isosceles triangle

Source: Sharygin Finals 2017, Problem 9.4

August 3, 2017

tangencygeometry

Problem Statement

Points

M

and

K

are chosen on lateral sides

AB,AC

of an isosceles triangle

ABC

and point

D

is chosen on

BC

such that

AMDK

is a parallelogram. Let the lines

MK

and

BC

meet at point

L

, and let

X,Y

be the intersection points of

AB,AC

with the perpendicular line from

D

to

BC

. Prove that the circle with center

L

and radius

LD

and the circumcircle of triangle

AXY

are tangent.

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