MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

64 Mathematical Properties
(a)
(b)
Figure 2.52: Propeller Theorem: (a) Symmetric Propellers. (b) Asymmetric
Propellers. (c) Triangular Hub. [138]
Property 68 (The Propeller Theorem). The Propeller Theorem states
that the midpoints of the three chords connecting three congruent equilateral
triangles which are joined at a vertex lie at the vertices of an equilateral triangle
(Figure 2.52(a)) [138].
In fact, the triangular propellers may even touch along an edge or overlap.
The Asymmetric Propeller Theorem states that the three equilateral triangles
need not be congruent (Figure 2.52(b)). The Generalized Asymmetric Propeller
Theorem states the propellers need not meet at a point but may meet
at the vertices of an equilateral triangle (Figure 2.52(c)). Finally, the General
Generalized Asymmetric Propeller Theorem states the the propellers need not
even be equilateral, as long as they are all similar triangles! If they do not
meet at a point then they must meet at the vertices of a fourth similar triangle
and the vertices of the triangular hub must meet the propellers at their
corresponding corners [138].
Figure 2.53: Tetrahedral Geodesics [305]
(c)