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As is well known, the centroid *G* of a triangle *ABC* divides, or equi-partitions, the triangle into three triangles, *AGB*, *BGC*, and *CGA*, of equal area as shown below.

What about a quadrilateral? Can we find a similar point *P* for a quadrilateral *ABCD* that divides, or equi-partitions, it into four triangles, *APB*, *BPC*, *CPD* and *DPA*, of equal area?

1) Use the dynamic sketch below to explore the problem by dragging *P* to see if you can find such a position.

2) Change the shape of *ABCD* to see for what kind of quadrilateral(s) you can find a position for *P* that satifies the 'equi-partitioning' condition stated above.

Equi-partitioning Point of a Quadrilateral

**Conjecture**

3) What conjectures have you made? Can you explain why (prove) that they are true?

4) Check your conjectures, and your proofs, in this paper by Shan & Poobhal Pillay (2010) at: *Equipartitioning and Balancing Points of Polygons*.

5) Also visit this webpage, which relates to the results mentioned in 4) above: More Properties of a Bisect-diagonal Quadrilateral.

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Release: 2020Q2, Semantic Version: 4.6.2, Build Number: 1047, Build Stamp: 139b185f240a/20200428221100

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Created by Michael de Villiers, 10 August 2020.