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.
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)
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
Angle Divider Theorem for a Cyclic Quadrilateral
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Created by Michael de Villiers, 1 Sept 2020 with WebSketchpad.