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Difficult geometry

Source: Own. Malaysian APMO CST 2024 P5

February 24, 2024

geometry

Problem Statement

Let

ABC

be a scalene triangle and

D

be the feet of altitude from

A

to

BC

. Let

I_1

,

I_2

be incenters of triangles

ABD

and

ACD

respectively, and let

H_1

,

H_2

be orthocenters of triangles

ABI_1

and

ACI_2

respectively. The circles

(AI_1H_1)

and

(AI_2H_2)

meet again at

X

. The lines

AH_1

and

XI_1

meet at

Y

, and the lines

AH_2

and

XI_2

meet at

Z

.
Suppose the external common tangents of circles

(BI_1H_1)

and

(CI_2H_2)

meet at

U

. Prove that

UY=UZ

.
Proposed by Ivan Chan Kai Chin

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