MathDB

Hidden devil properties of incircle and excircle

Source: Bulgaria JBMO TST 2023, Day 3, Problem 3

June 17, 2023

geometryincircleexcirclesimilaritycircumcircleincenter

Problem Statement

Let

ABC

be a non-isosceles triangle with circumcircle

k

, incenter

I

and

C

-excenter

I_C

. Let

M

be the midpoint of

AB

and

N

be the midpoint of arc

\widehat{ACB}

k

. Prove that

\angle IMI_C + \angle INI_C = 180^{\circ}