Explore
This sketch shows a quadrilateral that has been constructed so that its two sums of opposite sides are equal, i.e. a + b = c + d as shown.
1) Click on the Show Bisectors to show its angle bisectors. What do you notice about them?
2) Check you observation in 1) by clicking the Show Incircle button.
3) Change the shape of the quadrilateral by dragging X or Y, and repeat the above.
4) Formulate a conjecture.
Tangential Quadrilateral Converse
Challenge
1) Can you explain why (prove that) your conjecture above is true?
2) Hint: Try using proof by contradiction.
Explore More
To explore some more properties of this type of quadrilateral go to: Tangential Quadrilateral
Related Exploration
A similar dynamic sketch to the one above for a tangential quadrilateral can be made to illustrate the following important converse for cyclic quadrilaterals: If the two sums of the opposite angles of a quadrilateral ABCD are equal, i.e. ∠A + ∠C = ∠B + ∠D (= 180° if convex; = 360° if crossed), then the quadrilateral is inscribed in a circle (cyclic).
A classroom activity and guided proof of this converse cyclic quadrilateral result is given in the FREE DOWNLOAD of my book Rethinking Proof with Sketchpad (2012), pp. 82-84; 179-181. Download the associated Sketchpad sketches as a zipped file from Rethinking GSP 5 Sketches.
***************
Free Download of Geometer's Sketchpad
Back to "Dynamic Geometry Sketches"
Back to "Student Explorations"
Created by Michael de Villiers, 1 Sept 2020 with WebSketchpad; updated 7 April 2025.