A Property of 6-Parpolygon

Parpolygon is the term Howard Eves [Eves, pp. 232-235] employs to describe a 2n-gon with pairs of opposite sides parallel and equal. Following Kasner and Newman I call such polygons parapolygons, or in short paragons. I'll use the term parpolygon for a polygon with opposite sides parallel, but not necessarily equal.

Eves proves a property of 6-sided parpolygon: in any convex parpolygon, the triangles on odd and even vertices have equal areas.

The problem was offered at the 1958 Kürschák Prize Competition.

Solution

References

1. H. Eves, A Survey of Geometry, Allyn and Bacon, 1972
2. E. Kasner and J. Newman in their Mathematics and the Imagination, Dover Publications (March 28, 2001)

The right part of the applet area depicts two copies of the polygon with two triangles in question. If the "Hint" box is checked, the two right shapes appear to be dissected in three parallelograms and a small triangle. The sides of the latter are differences (in the absolute value) of the lengths of the opposite sides of the polygon and are, therefore, equal by SSS. The dissections show that each of the triangles have the area equal to average of the area of the polygon and the small triangle.

References

1. H. Eves, A Survey of Geometry, Allyn and Bacon, 1972