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APMO intersects on Luna Cabrera

Source: 2024 imocsl G4 (Boring Competiton P7)

August 8, 2024

geometryIMOC

Problem Statement

Given triangle

ABC

with

AB<AC

and its circumcircle

\Omega

. Let

I

be the incenter of

ABC

, and the feet from

I

to

BC

is

D

. The circle with center

A

and radius

AI

intersects

\Omega

at

E

and

F

.

P

is a point on

EF

such that

DP

is parallel to

AI

. Prove that

AP

and

MI

intersects on

AP \cap MO \in Luna

where

M

is the midpoint of arc

BAC

. [hide = Remark] In the test, the incenter called

Cabrera

and the circumcircle called

Luna

Cabrera

You have to prove

AP \cap MO \in Luna

Cabrera

Proposed by BlessingOfHeaven

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