Definition: An angle bisector of an angle is the locus (path) of all the points equidistant (same distance) from the two rays (or sides) forming the angle.
Explore
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?
(Hint: Use the idea of equidistance for all three challenges above).
To explore more properties of quadrilaterals with concurrent angle bisectors go to: Tangential Quadrilateral.
References
De Villiers, M. (1996). An Interesting Duality. Proceedings of the 2nd National Congress of AMESA, Peninsula Technikon, Cape Town, 1-5 July 1996, pp. 345-350.
De Villiers, M. (2020). The Tangential or Circumscribed Quadrilateral. Learning and Teaching Mathematics, No. 29, pp. 39-45.
Related Links
Water Supply: Four Towns (Rethinking Proof activity; equidistance)
Water Supply: Three Towns (Rethinking Proof activity; equidistance)
Exploring Rhombus Properties
Exploring Kite Properties
Logical Discovery: Tangential quadrilateral (Rethinking Proof activity, Pitot's theorem)
Parallelogram Angle Bisectors (Rethinking Proof activity)
Angle Bisectors of Trapezium Investigation
Angle Divider Theorem for a Cyclic Quadrilateral
Conway’s Circle Theorem as special case of Side Divider Theorem
Cyclic Quadrilateral Angle Bisectors Rectangle Result
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral
Bradley's Theorem, its Generalization & an Analogue Theorem
Visually Introducing & Classifying Quadrilaterals (Suggested Grades 1-7)
Exploring the Properties of (some) Quadrilaterals (Suggested Grades 4-9)
Constructing (some) Quadrilaterals (Suggested Grades 6-10)
Defining (some) Quadrilaterals (Suggested Grades 8-10)
Introducing, Classifying, Exploring, Constructing & Defining Quadrilaterals
A Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
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Created by Michael de Villiers, 1 Sept 2020 with WebSketchpad; updated 17 July 2026.