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equilateral triangle from Indian RMO

Source: CRMO 2015 Region 3 (West Bengal) p5

September 30, 2018

geometrycircumcircleincenterEquilateral Triangle

Problem Statement

Let

ABC

be a triangle with circumcircle

\Gamma

and incenter

I.

Let the internal angle bisectors of

\angle A,\angle B,\angle C

meet

\Gamma

in

A',B',C'

respectively. Let

B'C'

intersect

AA'

at

PIRQ

and

AC

in

Q.

Let

BB'

intersect

AC

in

R.

Suppose the quadrilateral

PIRQ

is a kite; that is,

IP=IR

and

QP=QR.

Prove that

ABC

is an equilateral triangle.

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