Asymmetric Propeller: What Is It?
A Mathematical Droodle

Explanation

Copyright © 1996-2009 Alexander Bogomolny

This is a generalization of a theorem about three equilateral triangles.

Vertices of three similar triangles are attached to the corresponsing vertices of the fourth triangle similar to the first three. All four have the same orientation. Pairs of unmatched vertices of the outside triangles are joint and the midpoints of those segments are connected. The triangle thus obtained is similar to the four given triangles. Here's a proof with complex numbers.

In the above diagram, I placed the origin O at one of the vertices and identified the straight line segments as complex numbers directed (unconventionally) from a (small yellow) square to a (small yellow) circle. X, Y, Z are the vertices of the "midpoint" triangle. The four given triangles are similar to a generic triangle with sides 1, k, and (1-k), where k is a complex number. The factors a, b, c, and d define the rotation and size of the triangles. We have

1. X = ((ka) + (a - c + kd))/2,
2. Y = (a - (1-k)c - kb)/2,
3. Z = ((a - c + d) + (a - (1-k)c + (1-k)b))/2,

from where we can compute the sides

4. YZ = Z - Y = (a + b - c + d)/2,
5. YX = X - Y = k(a + b - c + d)/2,
6. XZ = Z - X = (1-k)(a + b - c + d)/2.

So that triangle XYZ is indeed similar to the generic triangle (1, k, (1-k)).

References

1. American Mathematical Monthly, v 75, n 7, 1968, pp 732-739
2. L. Bankoff, P. Erdös and M. Klamkin, The Asymmetric Propeller, Mathematics Magazine, v. 46, n. 5, 1973, pp 270-272
3. M.Gardner, The Asymmetric Propeller, The College Mathematics Journal, v. 30, n. 1, 1999, (18-22)

Copyright © 1996-2009 Alexander Bogomolny