Concurrent Angle Bisectors of a Quadrilateral

Given a quadrilateral ABCD as shown below with the angle bisectors of angles A, B and C shown.
1) Drag any of the vertices A, B or C until the 3 angle bisectors are concurrent (meet in a point).
2) Click on the Show Bisector button to view the angle bisector of ∠D. What do you notice?
3) Click on the Show Incircle button to view the incircle to sides AB, BC and AD. What do you notice?
4) Change the shape of the quadrilateral, and repeat the above steps.
5) What conjecture(s) can you make?

Concurrent Angle Bisectors of a Quadrilateral

Challenge
1) Can you explain why (prove that) your conjecture above is true?
2) Can you generalize to any polygon?
3) Can you formulate & prove a similar result involving the perpendicular bisectors of the sides of a quadrilateral? And generalize to any polygon?

To explore more properties of this type of quadrilateral go to: Tangential Quadrilateral

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Created by Michael de Villiers, 1 Sept 2020 with WebSketchpad.