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Recommended Background Reading
The following two chapters from my Rethinking Proof book outlines the rationale for the book as well as the Van Hiele learning theory, which provided important guidance for the structuring & sequencing of the activities. However, these activities are not intended as a complete, self-contained geometry curriculum - these are more like snapshot suggestions for mathematics teachers to use and/or adapt as they see fit.
The Role and Function of Proof with Dynamic Geometry
The van Hiele Theory—Defining and Proving Within a Dynamic Geometry Context
Activities
The dynamic geometry activities below are from my book Rethinking Proof (free to download). As mentioned above these activities are intended as possible exemplars to introduce and illustrate different functions of proof to learners/students - see De Villiers (1990, 1999). To save valuable classroom time, many of the sketches are supplied ready-made in order to focus on the main tasks at hand, namely observing, experimenting, conjecturing & proving. However, some sketches require users to measure certain quantities, or to make some constructions or calculations, but Tools are provided in the WebSketchpad sketches below.
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Active links are underlined and shown in blue.
Explanation
Distances in an Equilateral Triangle (Viviani's theorem)
Water Supply I: Four Towns (Introducing perpendicular bisectors)
Water Supply II: Three Towns (Concurrency of perpendicular bisectors)
Triangle Angle Sum
Quadrilateral Angle Sum
Crossed Quadrilateral Angle Sum
Isosceles Trapezoid
Cyclic Quadrilateral
The Center of Gravity of a Triangle
Proof as Discovery
Kite Midpoints (Generalizes to Varignon's theorem & Orthodiagonal quadrilateral)
Logical Discovery (Varignon Parallelogram Perimeter)
Isosceles Trapezoid Midpoints
Logical Discovery: Circum Quad
Proof as Verification
Areas Ratios
Varignon Area
Logical Paradox
Cyclic Quadrilateral Converse
Concurrency Conjecture
Triangle Altitudes
Light Ray in a Triangle
Parallel Lines (Thomsen's Hexagon)
Proof as Challenge
Parallelogram Angle Bisectors
Parallelogram Squares
The Fermat-Torricelli Point
Airport Problem
Napoleon
Miquel
Proof as Systematization
Reasoning Backward: Triangle Midpoints
Reasoning Backward: Parallel Lines
Systematizing Rhombus Properties
Systematizing Isosceles Trapezoid Properties
References
De Villiers, M. (1990). The Role and Function of Proof in Mathematics. Pythagoras, 24, pp. 17-24.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Associated Sketchpad Sketches for Rethinking Proof
Rethinking Proof Sketches (zipped file)
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Published Reviews of Rethinking Proof
Published Reviews of Rethinking Proof
Back to "Dynamic Geometry Sketches"
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Michael de Villiers, created with WebSketchpad, 14 June 2025; updated 21 June 2025; 4/11/19/27 July 2025; 3 August 2025; 14 Sept 2025.