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Right Triangle with Incircle

Source: Romanian Masters in Mathematics 2020, Problem 1

March 1, 2020

geometryRMMRMM 2020

Problem Statement

Let

\omega

be a triangle with a right angle at

C

. Let

BC

be the incentre of triangle

ABC

, and let

AB

be the foot of the altitude from

C

to

AB

. The incircle

\omega

of triangle

ABC

is tangent to sides

C_1

,

CA

, and

AB

at

A_1

,

B_1

, and

C_1

, respectively. Let

E

and

F

be the reflections of

C

in lines

C_1A_1

and

C_1B_1

, respectively. Let

K

and

L

be the reflections of

D

in lines

C_1A_1

and

C_1B_1

, respectively.
Prove that the circumcircles of triangles

A_1EI

,

B_1FI

, and

C_1KL

have a common point.

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