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Lines from excentres are concurrent [ILL 1974]

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January 2, 2011

geometrycircumcircleincentergeometric transformationreflectiongeometry proposed

Problem Statement

Let

K_a,K_b,K_c

with centres

O_a,O_b,O_c

be the excircles of a triangle

ABC

, touching the interiors of the sides

BC,CA,AB

at points

T_a,T_b,T_c

respectively. Prove that the lines

O_aT_a,O_bT_b,O_cT_c

are concurrent in a point

P

for which

PO_a=PO_b=PO_c=2R

holds, where

R

denotes the circumradius of

ABC

. Also prove that the circumcentre

O

of

ABC

is the midpoint of the segment

PI

, where

I

is the incentre of

ABC

.

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