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Triple product of three sides equals another triple product

Source: Benelux MO 2013 Q3

April 29, 2013

geometrycircumcircleratiopower of a pointgeometry unsolved

Problem Statement

Let

\triangle ABC

be a triangle with circumcircle

\Gamma

, and let

I

be the center of the incircle of

\triangle ABC

. The lines

AI

,

BI

and

CI

intersect

\Gamma

in

D \ne A

,

E \ne B

and

F \ne C

. The tangent lines to

\Gamma

in

F

,

D

and

E

intersect the lines

AI

,

BI

and

CI

in

R

,

S

and

T

, respectively. Prove that

\vert AR\vert \cdot \vert BS\vert \cdot \vert CT\vert = \vert ID\vert \cdot \vert IE\vert \cdot \vert IF\vert.

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