MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Properties 51
Figure 2.30: Equilateral Shadows [180]
Property 40 (Equilateral Shadows). Any triangle can be orthogonally projected
onto an equilateral triangle [180]. Moreover, under the inverse of this
transformation, the incircle of the equilateral triangle is mapped to the “midpoint
ellipse” of the original triangle with center at the triangle centroid and
tangent to the triangle sides at their midpoints (Figure 2.30).
Note that this demonstrates that if we cut a triangle from a piece of paper
and hold it under the noonday sun then we can always position the triangle
so that its shadow is an equilateral triangle.
Figure 2.31: Fundamental Theorem of Affine Geometry [33]
Property 41 (Affine Geometry). All triangles are affine-congruent [33]. In
particular, any triangle may be affinely mapped onto any equilateral triangle
(Figure 2.31).
This theorem is of fundamental importance in the theory of Riemann surfaces.
E.g. [288, p. 113]:
Theorem 2.1 (Riemann Surfaces). If an arbitrary manifold M is given
which is both triangulable and orientable then it is possible to define an analytic
structure on M which makes it into a Riemann surface.