A Final Chapter of the Asymmetric Propeller Story

What is this about?

12 July 2013, Created with GeoGebra

Problem

Given three similar triangles $A_{1}B_{1}C_{1},$ $A_{2}B_{2}C_{2},$ $A_{3}B_{3}C_{3}.$ The centroids of the triangles $A_{1}A_{2}A_{3},$ $B_{1}B_{2}B_{3},$ and $C_{1}C_{2}C_{3}$ form a fourth similar triangle

Given three similar triangles A1B1C1, A2B2C2, A3B3C3. The centroids of the triangles A1A2A3, B1B2B3, and C1C2C3 form a fourth similar triangle

Hint

I am afraid this is going to be either vectors or complex numbers. Hopefully, using spiral similarities will simplify the proof.

Solution

Let $S$ and $T$ be spiral similarities that map $A_{1}B_{1}C_{1}$ onto $A_{2}B_{2}C_{2}$ and $A_{3}B_{3}C_{3},$ respectively:

$S(A_{1}B_{1}C_{1})=A_{2}B_{2}C_{2}$ and
$T(A_{1}B_{1}C_{1})=A_{3}B_{3}C_{3}.$

Define $A=(A_{1}+A_{2}+A_{3})/3 = (A_{1}+S(A_{1})+T(A_{1}))/3.$ The affine operator $M = (I+S+T)/3,$ where $I$ is the identity operator, preserves angles such that if $B=M(B_{1})$ and $C=M(C_{1})$ triangle $ABC$ is similar to $\Delta A_{1}B_{1}C_{1}.$

That's all folks.

Acknowledgment

The problem of the Asymmetric Propeller has a long history. The original problem underwent several generalizations associated with the names of Martin Gardner, Leon Bankoff, Paul Erdös, Murrey Klamkin, and Ross Honsberger. The problem relates to the Fundamental Theorem of 3-Bar Motion discovered by William Kingdom Clifford and Arthur Cayley.

Most recently, Dao Thanh Oai has posted at the CutTheKnotMath facebook page a special case of the above statement where the three given triangles were equilateral. The post brought up the memory of the Asymmetric Propeller which led to the more general case of three similar triangles, as stated above. In all likelihood, this is the most natural environment for the problem.

Asymmetric Propeller

  1. Asymmetric Propeller (An Interactive Gizmo)
  2. Asymmetric Propeller: a Generalization
  3. A Case of Similarity
  4. Napoleon's Propeller
  5. Asymmetric Propeller and Napoleon's Theorem
  6. Asymmetric Propeller by Plane Tiling
  7. A Final Chapter of the Asymmetric Propeller Story

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