One
seventh of a triangle
The sides of a triangle are divided into thirds. Segments are
constructed from one of the points marking a third on each side with
the opposite vertex as shown below. A small triangle is formed at the
center.
In the hinged figure, the midpoints of the sides of the triangle play
an important role. To see what the area of the blue triangle is,
additional segments are constructed connecting midpoints to the mark of
the other third. For example, E is a midpoint and G marks the other
third on the base.
In order to prove the identities suggested by the hinged figure, it is
necessary to show that segment EG goes through the vertex of the blue
triangle, that is, that points EFG are in a straight line. This can be
done using coordinates.
Once this is done, it is easy to see that the rotated small yellow
triangles will indeed form complete parallelograms on the sides of the
blue triangle. The figures below show one triangle rotated 180˚ around
the midpoint E. The parallelogram has twice the area as the blue
triangle.
