AoPS - Problem

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Problem

Source: MOP 2005 Homework - Blue Group #20

Tags: geometry, circumcircle, incenter, exterior angle, angle bisector, geometry unsolved

Konigsberg

18.05.2014 19:25

A circle which is tangent to sides $AB$ and $BC$ of triangle $ABC$ is also tangent to its circumcircle at point $T$ . If $I$ in the incenter of triangle $ABC$ , show that $\angle ATI=\angle CTI$ .

pi37

18.05.2014 19:49

This is equivalent to showing that $T,I,M$ are collinear, where $M$ is the midpoint of arc $ABC$ . Consider an inversion about $B$ . This maps $T'$ to the excircle touch-point on $A'C'$ , $I'$ to the $B$ -excenter of $BA'C'$ , and $M'$ to the intersection of the exterior angle bisector of $A'BC'$ and $A'C'$ . It is sufficient to show $B,T',I',M'$ are concyclic, but this is clear as $M'BI'=M'T'I'=\pi$ .

jayme

19.05.2014 08:22

Dear Mathlinkers, see O.M. Turquie (2003) or http://perso.orange.fr/jl.ayme vol. 4 A new mixtilinear incircle adventure I p. 18. Sincerely Jean-Louis

Mate007

30.01.2018 18:11

See it's the the general property is mixtilinear in circle that $T,I,M$ are collinear. And then inversion about $B$ gives rest of the thing as done by pi37

Drunken_Master

31.01.2018 16:42

@above I think he knows this fact after 3.5 years.