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Fair Sharing of an Equilateral Triangular Pizza

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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.
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