Asymmetric Propeller: a Generalization

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

Buy this applet
What if applet does not run?

Explanation

Buy this applet
What if applet does not run?

This is a variant of the Asymmetric Propeller theorem, a simple generalization and the last one to deal with three equilateral triangles. The applet attempts to illustrate the proof by L. Bankoff, P. Erdös and M. S. Klamkin.

The three equilateral triangles with a shared vertex O may be conveniently denoted as OAA', OBB' and OCC'. Two triangles ABC and A'B'C' are equal, similarly oriented and are obtained from each other by a rotation through 60^o. If O is the center of a coordinate system and d a rotation through 60^o in the positive direction, then A' = dA, B' = dB, and C' = dC. For j = d², a rotation through 120^o, a well-known criterion is used to verify whether a triangle is equilateral or not. In our case, we should check whether

j²(A + B') + j(B + C') + (C + A') = 0.

Let's see. Since j² = d⁴ = -d,

j²(A + B') + j(B + C') + (C + A')	= -d(A + dB) + d²(B + dC) + (C + dA)
	= -dA - d²B + d²B - C + C + dA
	= 0.

So indeed, the triangle in question is equilateral.

Hubert Shutrick found an elegant geometric argument that could be described as a "nondiscrete induction": if the statement of the theorem holds for one configuration of three equilateral triangles then it also holds for a modification of the latter. Since one can always start with a symmetric configuration of three congruent equilateral triangles for which the statement obviously holds, it holds for any configuration.

Asymmetric Propeller

Asymmetric Propeller (An Interactive Gizmo)
Asymmetric Propeller: a Generalization
A Case of Similarity
Napoleon's Propeller
Asymmetric Propeller and Napoleon's Theorem
Asymmetric Propeller by Plane Tiling

40608141