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turkey nmo 2006 q5

Source:

January 6, 2007

geometryincentercircumcirclegeometric transformationhomothetytrigonometrygeometry unsolved

Problem Statement

ABC

be a triangle. Its incircle touches the sides

CB, AC, AB

respectively at

N_{A},N_{B},N_{C}

. The orthic triangle of

ABC

is

H_{A}H_{B}H_{C}

with

H_{A}, H_{B}, H_{C}

are respectively on

BC, AC, AB

. The incenter of

AH_{C}H_{B}

is

I_{A}

;

I_{B}

and

I_{C}

were defined similarly. Prove that the hexagon

I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}

has all sides equal.

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