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isosceles criterion with touchpoints with circle (Kyiv City Olympiad 2005 11.2)

Source:

June 27, 2020

geometryisoscelescircle

Problem Statement

A circle touches the sides

AC

and

AB

of the triangle

ABC

at the points

{{B}_ {1}}

and

{{C}_ {1}}

respectively. The segments

B {{B} _ {1}}

and

C {{C} _ {1}}

are equal. Prove that the triangle

ABC

is isosceles.
(Timoshkevich Taras)