# Fair Sharing of an Equilateral Triangular Pizza

Fair Sharing of an Equilateral Triangular Pizza

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: ≈0.866025 units.

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PA

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PA'

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3

2

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.

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