A Surprising Constant Area Sum Involving Translating Figures

The following surprising constant area sum result by James Tanton and Brad Ballinger was brought to our attention by Daniel Scher who self-professes that he is “addicted to triangle shearing problems” - see A Beautiful Application of Shearing (Scher, 2025).

Explore
An arbitrary △𝐷𝐸𝐹 sits inside a larger arbitrary △𝐴𝐵𝐶.
1) Click on and drag the 'interior' (not the vertices) of △𝐷𝐸𝐹.
(In other words, translate △𝐷𝐸𝐹 without any rotation or change of shape).
2) What do you notice about the sum of the areas of the three green triangles?
3) Change the shape of either one or both of the triangles & repeat step 1).
4) Is the result still true if △𝐷𝐸𝐹 is dragged anywhere, far outside △𝐴𝐵𝐶?

Challenge
5) Can you explain why (prove that) your observations in 1-4) are true?
Proof Hints
a) Click on the 'Proof Hint' button to construct a parallel to 𝐷𝐸 through 𝐴, a parallel to 𝐸𝐹 through 𝐵, and a parallel to 𝐹𝐷 through 𝐶, yielding △𝑃𝑄𝑅.
b) Then consider what happens if we move the vertices A and B of the green triangles to Q, etc.
c) Use the concept of 'directed areas' (see below, Coxeter & Greitzer, 1967) to prove your observation in 4).

Surprising Constant Area Sum Involving Translating Figures

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6) Click on the 'Link to Quadrilateral Area Sum Problem' button to navigate to a new sketch.
7) Follow the instructions in the new sketch.
8) What do you notice about the sum of the areas of the four triangles?
9) Does the result also hold if EFGH is dragged anywhere, far outside ABCD?
10) Challenge: Can you explain why (prove that) the result also holds for a quadrilateral, even when EFGH is anywhere outside?
Hint: Use the concept of 'directed areas' (see below, Coxeter & Greitzer, 1967).
11) Does the result hold if either one or both of EFGH and ABCD are concave or crossed?
12) Can you generalize further?
13) Can you think of a way to generalize the triangle result to an analogous 3D result?
14) Challenge: Can you prove your generalizations in 12) and 13)?

Submitted Paper
A paper "A Surprising Constant Sum When Translating Figures in Figures" by Hans Humenberger & myself about the above results has been submitted for consideration for publication. All rights reserved.

Some Related References
Coxeter, H. S. M. & Greitzer, S. L. (1967). Directed Areas Excerpt from Geometry Revisited. MAA, Washington, pp. 51-54.
De Villiers, M. (1993). Transformations: A golden thread in school mathematics. Spectrum, 31(4), 11-18.
De Villiers, M. (2014). Slaying a Geometrical Monster: Finding the Area of a Crossed Quadrilateral. Scottish Mathematical Council Journal, 44 (Dec), 71-74.
De Villiers, M. (2020). The Value of using Signed Quantities in Geometry. Learning and Teaching Mathematics, No. 29, pp. 30-34.
Ng, O-L. & Sinclair, N. (2015). “Area Without Numbers”: Using Touchscreen Dynamic Geometry to Reason About Shape. Canadian Journal of Science, Mathematics & Technology Education, 15(1), 84–101.

External Links
Some Triangle Shearing Investigations (Sine of the Times)
A Beautiful Application of Shearing (Sine of the Times)
Shear magic (NRICH - University of Cambridge)
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, 30 Sept 2025; updated 2-3 Oct 2025.