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Beautiful conditional geo in China Southeast

Source: 2023 China Southeast Grade 11 P3

July 30, 2023

geometryangle bisector

Problem Statement

In

\triangle {ABC}

,

{D}

is on the internal angle bisector of

\angle BAC

and

\angle ADB=\angle ACD

.

E, F

is on the external angle bisector of

\angle BAC

, such that

AE=BE

and

AF=CF

. The circumcircles of

\triangle ACE

and

\triangle ABF

intersects at

{A}

and

{K}

and

A'

is the reflection of

{A}

with respect to

BC

. Prove that: if

AD=BC

, then the circumcenter of

\triangle AKA'

is on line

AD

.

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