“Finding Triangles of Equal Areas in a Convex Quadrilateral”

by

Christopher R. Whitworth

You have been
given a single convex quadrilateral.

I will be
proving that even in this convex quadrilateral, you can find triangles of equal
areas. The first step in finding
four triangles of equal areas is to find the midpoint on each line of the
convex quadrilateral.

Next, you will connect the
midpoints with segments as shown below.
Notice that the figure inside of the convex quadrilateral appears to be
a parallelogram.

A way to check to see if
the figure inside of the convex quadrilateral is indeed a parallelogram is to
create line segments to and from each point in the figure and to create lines
that are parallel to the lines in the figure (Hint: The parallel lines should pass through the midpoints of each
line segment on the edge of the figure).
If the segments meet at one point in the center of the figure, then the
figure is a parallelogram.

The
segments do indeed meet in the center of the figure. Therefore, it is a parallelogram.

The next step is to create
another parallelogram inside of the first parallelogram by making its vertexes
meet one the midpoints of the first parallelogram.

Next, highlight the four
triangles that was created by the second parallelogram being placed inside of
the first parallelogram.

Then,
you would like to measure the area of each triangle by either using Geometers
Sketchpad 4.0 or by using the formula, A = L x W.

Notice that the areas of
each of the four triangles are equal.
The four triangles are not congruent, but are equal in area. If the convex quadrilateral is moved
into a different position, the areas of the four triangles should still be
equal to one another. If they do
remain equal to one another, then it is possible to create triangles that are
equal in area inside of a convex quadrilateral.

The convex quadrilateral
was moved into a different position.
Did the triangles within the parallelogram that is inside of the
quadrilateral remain equal to each other? Yes. I have
proven that you can find four triangles that are equal in area, but not
congruent to each other, inside of a convex quadrilateral.

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