About 2004, a 14 year old boy David Cross in the UK (re)discovered the interesting result below and it's now often referred to as Cross's Theorem:

"Given any triangle *ABC* with squares *ABED*, *BCGF* and *CSIH* constructed on the sides, then the areas of triangles *ABC*, *EBF*, *GCH* and *IAD* are equal."

However, apparently the result is historically attributed much earlier to a 19th-century French mathematician, Vecten, who extensively studied various properties of this configuration.

Cross's (Vecten's) theorem & generalizations to quadrilaterals

Note that the theorem is also true if the squares are constructed inwards. For example, drag *B* across *AC* to dynamically view such a configuration.

**Challenge**

1) Try and *explain why* (prove that) the result is true. If so, can you explain/prove it in different ways?

2) But if you get stuck, have a look at the (unedited) proof of one of my 2007 Kennesaw graduate students at *Gin Chou's proof*.

**Generalize**

1) Can you generalize the result to quadrilaterals?

2) Explore the analogous cases for quadrilaterals below, and also try to explain why (prove) they are true.

3) What happens if one continues constructing squares on the triangular 'flanks' formed by the squares on the sides of a triangle? See (e) below.

**Investigate**

In the sketch above, navigate to each of the cases below by clicking on the appropriate buttons to explore:

a) Similar rectangles on sides of quadrilateral

b) Similar parallelograms on sides of quadrilateral 1

c) Similar parallelograms on sides of quadrilateral 2

d) Similar cyclic quadrilaterals on sides of quadrilateral

e) Continuing constructing squares on the 'flanks' of a Cross triangle

f) Continuing constructing squares on the 'flanks' of a Cross quadrilateral

Regarding a-d) above, read my 2007 article An Example of the Discovery Function of Proof in *Mathematics in School*.

Regarding e) above, consult Mason, J. (2008). Flanking Figures. *Mathematics in School*, 37(1), pp. 28-29.

A copy of various reader responses to the posing by Geoff Faux in the UK journal *Mathematics Teaching*, 189 in (2004) of Cross's Theorem as a problem, can be found at Responses.

For more reading on the above topics, consult this 2018 article Modification Cross' Theorem on Triangle with Congruence by Syawaludin et al in the *International Journal of Theoretical and Applied Mathematics*.

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Created by Michael de Villiers, 27 May 2010; updated 11 November 2020 with *WebSketchpad*.