Triangulated Tangential Hexagon theorem

Triangulated Tangential Hexagon theorem
If a tangential hexagon ABCDEF is triangulated by drawing 3 diagonals from any of its vertices, and the incircles of the four formed triangles are constructed, then the distance between the two tangent points of the incircles to the 1st diagonal is equal to the distance between the two tangent points of the incircles to the 3rd diagonal.

Triangulated Tangential Hexagon theorem

Challenge
1) Can you explain why (prove that) this result is true?
2) Can you generalize further to other tangential polygons?

Hint: If stuck, try applying the results in these two webpages Theorem of Gusić & Mladinić and Tangential/ Circumscribed Hexagon.

Read my paper The Tangential or Circumscribed Quadrilateral in Learning & Teaching Mathematics, Dec 2020.

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Created by Michael de Villiers, 2 Sept 2020 with WebSketchpad, updated 5 March 2021.