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Too many perpendicular and parallel lines

Source: Sharygin 2022 P20

March 4, 2022

geometry

Problem Statement

Let

O

,

I

be the circumcenter and the incenter of

\triangle ABC

;

R

,

r

be the circumradius and the inradius;

D

be the touching point of the incircle with

BC

; and

N

be an arbitrary point of segment

ID

. The perpendicular to

ID

at

N

meets the circumcircle of

ABC

at points

X

and

Y

. Let

O_{1}

be the circumcircle of

\triangle XIY

.
Find the product

OO_{1}\cdot IN

.

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