Kite Midpoints

The dynamic geometry activities below are from the "Proof as Discovery" 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: Kite Midpoints Worksheet & Teacher Notes.

Prerequisites
It is assumed here that you already know all the properties of a kite, and specifically those of its diagonals. If not, please first complete the following two activities:
Visually Introducing & Classifying a Kite
Exploring Kite Properties

Kite Midpoints

Notes
1) This is the first activity in Rethinking Proof which focusses on the discovery function of proof; that is, to show how by explaining (proving) something and identifying its underlying characteristic property, we can sometimes immediately generalize the result.
2) In this particular activity, students and learners are encouraged to reflect carefully on the guided proof, and then generalize the result in two different ways:
.....a) to a general quadrilateral (Varignon's theorem)
.....b) to an orthodiagonal quadrilateral (a quadrilateral with perpendicular diagonals).
3) Ensure that you check all your conjectures above by sufficient dragging, also checking concave as well as crossed cases.
4) Except for the last page, users have to make their own constructions in the first three sketches using the tools on left (scroll down if necessary).
5) Click on the 'Link to area orthodiagonal quad' button to navigate to a new sketch measuring the area of an orthodiagonal quadrilateral.
6) What conjecture can you make regarding the relationship between the diagonals and its area?
7) Is it still valid if the orthodiagonal quadrilateral becomes concave or crossed?
8) Challenge: Can you explain why (prove that) your conjecture in 6) & 7) is true?
9) As shown in Hanna & Jahnke (2002), one can also use arguments from physics to explain/prove Varignon's theorem.

References
De Villiers, M. (1990). The Role and Function of Proof in Mathematics. Pythagoras, No. 24, pp. 17–24.
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry (free to download). Dynamic Mathematics Learning.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Hanna, G. & Jahnke, H.N. (2002). Arguments from Physics in Mathematical Proofs: An Educational Perspective. For the Learning of Mathematics, Vol. 22, No. 3 (Nov.), pp. 38-45.
Oliver, P.N. (2001). Pierre Varignon and the Parallelogram Theorem. Mathematics Teacher, Vol. 94, No. 4, April, pp. 316-319.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Varignon's theorem (Wikipedia)
Orthodiagonal quadrilateral (Wikipedia)
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, 19/27 July 2025; updated 18 Nov 2025.