# Fair Sharing of an Equilateral Triangular Pizza

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This Demonstration sets in motion the lovely proof without words given in [1] of a version of the pizza theorem. Drag the marked point to change the image.

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Contributed by: Chris Boucher (July 2016)

After work by: Grégoire Nicollier

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

It appears that this theorem can be generalized somewhat by allowing the marked point to stray outside the triangle and computing signed areas. To compute the signed area of a triangle, assign an order to the vertices and take the area to be positive if this ordering traces the border of the triangle in a counterclockwise fashion and negative otherwise. Order the vertices of the three blue and three white triangles so that each triangle has positive signed area while the marked point is in the interior of the triangle. With this arrangement, it appears that the blue and white signed areas are the same whether the marked point is inside or outside the big triangle.

Reference

[1] G. Nicollier, "Proof without Words: Half Issues in the Equilateral Triangle and Fair Pizza Sharing," *Mathematics Magazine*, 88(5), 2015 p. 337.

## Permanent Citation