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Show that (IDP) and (IHX) are tangent to each other

Source: IMOC 2021 G10

August 11, 2021

geometrytangent circlesCircumcenterincenter

Problem Statement

Let

O

,

I

be the circumcenter and the incenter of triangle

ABC

, respectively, and let the incircle tangents

BC

at

D

. Furthermore, suppose that

H

is the orthocenter of triangle

BIC

,

N

is the midpoint of the arc

BAC

, and

X

is the intersection of

OI

and

NH

. If

P

is the reflection of

A

with respect to

OI

, show that

\odot(IDP)

and

\odot(IHX)

are tangent to each other.

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