The dynamic geometry activities below are from the "Proof as Challenge" 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: Parallelogram Angle Bisectors Worksheet & Teacher Notes.
Angle Bisectors of some Quadrilaterals
Note
For the general quadrilateral case, ensure that you also drag ABCD into a concave as well as a crossed shape.
Note: When ABCD is concave or crossed, EFGH becomes a crossed cyclic quadrilateral, so the proof needs to be adapted using directed angles and requires knowledge of the properties of crossed quadrilaterals - for a proof, see De Villiers (1994, 191–192).
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. (2020). The Value of using Signed Quantities in Geometry . Learning and Teaching Mathematics, No. 29, pp. 30-34.
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Related Links
The Perpendicular Bisectors of a Parallelogram
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral
The quasi-incentre of a quadrilateral
Equi-inclined Lines to the Sides of a Quadrilateral at its Vertices
The Equi-inclined Bisectors of a Cyclic Quadrilateral
Concurrent Angle Bisectors of a Quadrilateral
Parallelogram Distances
Cyclic Quadrilateral Angle Bisectors Rectangle Result
Cyclic Quadrilateral Incentres Rectangle (Japanese theorem for cyclic quadrilaterals)
Angle Divider Theorem for a Cyclic Quadrilateral
Pitot's Theorem for a tangential quadrilateral
Equi-inclined Lines Problem
Crossed Quadrilateral Properties
External Links
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, 9 June 2025.