The dynamic geometry activities below are from the "Proof as Verification" section of my book Rethinking Proof (free to download).
Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Cyclic Quadrilateral Converse Worksheet & Teacher Notes.
Prerequisites
1) Though not essential, it is suggested to first complete the activity at Water Supply: Four Towns
2) and highly recommended to have already engaged with Cyclic Quadrilateral activity before completing the activity below.
Cyclic Quadrilateral Alternate Angles Converse
Notes
1) Since it is difficult to drag the points B and D so that the sum of the associated opposite angles is precisely 180°, this worksheet focuses on further elaborating the verification function of a proof. In other words, that proof can be used to convince or to remove lingering doubts.
2) Follow the guided proof in the accompanying Worksheet.
3) For a more accurate and convincing sketch that dynamically illustrates the truth of the converse, click on the 'Link to Cyclic Quad Converse 2' button.
4) In the dynamic sketch in 3), a quadrilateral EFGH has been constructed with opposite (alternate) angles HEF and HGF supplementary by constructing them respectively equal to angles DCA and DCB lying adjacent to each other on a straight line. This sketch shows that the perpendicular bisectors are always concurrent and that a circumcircle always passes through all four vertices.
5) What about the converse for the cyclic hexagon case at: Cyclic Hexagon Alternate Angles Sum? Is it also true? Can you prove or disprove it?
6) Check your answer to 5) by consulting De Villiers (2006).
7) A study by Leung (2002) also describes ‘Cabri proof by contradiction’ of the converse 'alternate (opposite) angle' theorem about cyclic quadrilaterals given by a pair of 16 year-old students in a Hong Kong secondary school. This is a recommended reading.
References
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry (free to download). Lulu Press.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2006). Recycling Cyclic Polygons Dynamically. Mathematics in School, Vol. 35, no. 3, pp. 2-4.
Leung, A. (2002). Theorem Justification and Acquisition in Dynamic Geometry: A Case of Proof by Contradiction. International Journal of Computers for Mathematical Learning, 7(2):145-165. DOI:10.1023/A:1021195015288
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Some Related Links
Water Supply: Four Towns (Rethinking Proof activity)
Cyclic Quadrilateral Aternate Angles Sum (Rethinking Proof activity, alternate angles supplementary)
Cyclic Hexagon Alternate Angles Sum (Generalizes above)
Visually Introducing & Classifying Quadrilaterals by Dragging (Grades 1-7)
A generalization of the Cyclic Quadrilateral Alternate Angle Sum theorem
Parallelogram Angle Bisectors
Crossed Quadrilateral Properties
The Fermat-Torricelli Point (Rethinking Proof activity)
Airport Problem (Rethinking Proof activity)
Napoleon's Theorem (Rethinking Proof activity)
Miquel's Theorem (Rethinking Proof activity)
Triangle Altitudes (Rethinking Proof activity)
Light Ray in a Triangle (Rethinking Proof activity - Fagnano)
Systematizing Isosceles Trapezoid Properties - Defining an Isosceles Trapezoid (Rethinking Proof activity)
Some Properties of Bicentric Isosceles Trapezia & Kites
Angle Divider Theorem for a Cyclic Quadrilateral
Conway’s Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral
The Equi-inclined Bisectors of a Cyclic Quadrilateral
Two British Mathematics Olympiad Concurrency Problems
A variation of Miquel's theorem and its generalization
Similar Parallelograms: A Generalization of a Golden Rectangle property
Japanese theorem for cyclic quadrilaterals
Cyclic Quadrilateral Angle Bisectors Rectangle Result
Cyclic Kepler Quadrilateral Conjectures
Some Circle Concurrency Theorems
The quasi-circumcentre of a quadrilateral
Four Collinear Points
External Links
Cyclic quadrilateral (Wikipedia)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, created with WebSketchpad, 2 July 2025; updated 13 Jan 2026; 2 April 2026; 15 July 2026.