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?