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.


The theorem states: Let be a point in an equilateral triangle . Construct six segments: , , and from to the vertices of and the perpendiculars , , and from to the sides of . The segments divide into six triangles, colored alternately white and blue. Then the blue and white regions have equal areas: units.

Put another way, if the triangle were a pizza, it would be fairly shared by two people who took alternate slices around the marked point.

Clicking "lines" draws dashed lines through parallel to the sides, dividing into three parallelograms and three isosceles triangles, each of which is evenly divided by the blue and white regions. Clicking "parallelograms" or "triangles" colors these smaller regions.


Contributed by: Chris Boucher (July 2016)
After work by: Grégoire Nicollier
Open content licensed under CC BY-NC-SA



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.


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

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