Segments between tangency points on opposite sides of a quadrilateral | geometry | intermediate

Episode 81.

Segments between tangency points on opposite sides of a quadrilateral | geometry | intermediate.
The segments between the tangency points of the incircles of the triangles formed by the sides and the diagonals of a quadrilateral | plane geometry | intermediate level.

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

Theorem. Let $ABCD$ be an arbitrary quadrilateral. We consider the incircles of the triangles $ABC$, $BCD$, $CDA$, $DAB$. Then the segments between the tangency points of these circles on the sides $AB$ and $CD$ are equal to each other. Also, the segments between the tangency points of these circles on the sides $BC$ and $DA$ are equal to each other.

Mathematics. Geometry. Plane geometry.
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The same video on YouTube:
https://youtu.be/lwRsY-qUpHM