Fair Sharing of an Equilateral Triangular Pizza

lines

parallelograms

triangles

labels

●

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

P

be a point in an equilateral triangle

ABC

. Construct six segments:

PA

,

PB

, and

PC

from

P

to the vertices of

ABC

and the perpendiculars

PA'

,

PB'

, and

PC'

from

P

to the sides of

ABC

. The segments divide

ABC

into six triangles, colored alternately white and blue. Then the blue and white regions have equal areas:

3

2

≈0.866025

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

P

parallel to the sides, dividing

ABC

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.