Tangent lengths to the incircle and the excircle | plane geometry | intermediate level

Episode 64.

Tangent lengths to the incircle and the excircle | plane geometry | intermediate level.

Branch of mathematics: plane geometry.
Difficulty level: intermediate.

Theorem. Let $ABC$ be a triangle. Let $T_A$, $T_B$, $T_C$ be the tangency points of the incircle of the triangle $ABC$ with the sides $BC$, $CA$, $AB$ respectively. Let $T_A'$, $T_B'$, $T_C'$ be the tangency points of the excircle of the trinagle $ABC$ with the extension of the side $BC$, the side $CA$, and the extension of the side $AB$ respectively. Then $AT_B=CT_B'$ and $AT_B'=CT_B$.

Mathematics. Geometry. Plane geometry.
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