The asymmetric propeller

Statement of the problem

The three arbitrary-sized equilateral triangles $T_1, T_2, T_3$ share a vertex $O$, as seen in the diagram above. Their remaining vertices are connected to each other with the dotted lines as shown. Let $P$, $Q$, $R$ be the midpoints of the connecting lines.

Proposition 1: The triangle $PQR$ is equilateral.

I learned about this problem from Martin Gardner's article [1]; see the references below. Read it, if you can, for interesting background details.

A generalization

Gardner writes that he learned about this problem from Leon Bankoff, a mathematically inclined dentist! Bankoff, Erdos and Klamkin had given two different proofs of Proposition 1 in their 1973 article [2]. By the time Bankoff and Gardner met in 1979, Bankoff had generalized the previous result in several ways (but never published it.) The diagram below shows one of the generalizations. Here, the equilateral triangles have been replaced by arbitrary triangles, and their common point $O$ has been replaced by the vertices of a triangle $ABC$. Assume the three outer triangles and $ABC$ are similar, and that their common corners with $ABC$ are at the corresponding angles. Join their outer vertices as before and locate the midpoints $P$, $Q$, $R$ of the joining segments.

Proposition 2: The triangle $PQR$ is similar to the triangle $ABC$.

See Gardner's article for a proof.

Two additional comments

• In [1] Gardner raises the question whether the triangles may be replaced by other polygons. Squares, perhaps. The article [3], written by three undergraduate students, answers the question in the negative—apparently their various attempts to generalize Bankoff's constructions to squares did not succeed.
• An extensive interview with Bankoff appears in [4]. Dr. Bankoff died in 1997.

References

1. Martin Gardner, The Asymmetric Propeller, The College Mathematics Journal, vol. 30, no. 1, January 1999, pp. 18–22.
2. Leon Bankoff, Paul Erdos, Murray S. Klamkin, The Asymmetric Propeller, Mathematics Magazine, vol. 46, no. 5, November 1973, pp. 270–272.
3. Gillian Saenz, Christopher Jackson, Ryan Crumley, The Asymmetric Propeller revisited, The College Mathematics Journal, vol. 31, no. 5, November 2000, pp. 347–349.
4. G. L. Alexanderson and Leon Bankoff, A Conversation with Leon Bankoff, The College Mathematics Journal, vol. 23, no. 2, March 1992, pp. 98–117.

This page was created on July 15, 2002 and was last updated in February 2020.