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NPC lies on the A-Napoleon line of another triangle

Source: 2017 Taiwan TST Round 3, Test 1, Problem 3

April 26, 2017

geometrygeometry proposed

Problem Statement

Given a

\triangle ABC

with circumcircle

\Gamma.

Let

A'

be the antipode of

A

in

\Gamma

and

D

be the point s.t.

\triangle BCD

is an equilateral triangle (

A

and

D

are on the opposite side of

BC

). Let the perpendicular from

A'

to

A'D

cuts

CA,

AB

at

E,

F,

resp. and

T

be the point s.t.

\triangle ETF

is an isosceles triangle with base

EF

and base angle

30^{\circ}

(

A

and

T

are on the opposite side of

EF

). Prove that

AT

passes through the 9-point center of

\triangle ABC.

Proposed by Telv Cohl

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