“Finding Triangles of Equal Areas in a Convex Quadrilateral”


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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