## A regular triangle and an anti-triangle

In this interactive diagram are a red circle and within it a red equilateral triangle, labeled s0, s1 and s2, and also a blue circle and triangle, labeled a0, a2 and a1. As in the previous diagram there are also blue and red dots to the right of the circles which control their radii. The point marked O at the middle is a fixed origin for the diagram. Note the actual labels in the diagram are black, for I can't color them with GSP: the colors are associated to the triangles, circles and lines.

Note that the orientation of the triangle s (for Standard) is postive (counter-clockwise), and that of triangle a for Anti-standard, is negative (clockwise).

As the diagram opens you see also a single green point labeled r2 on the line from a2 to s2. The labels are initially hard to read because a0 is on top of s0, a2 is on top of s1, and a1 on top of s2.

To see better drag the point a0 part of the way round its blue circle. The result will be the appearance of two more green points r1 and r0 joined to r2 by a straight line through O. Each r point is midway between the corresponding s and a points. To draw the lines in green of which we have the midpoints press [Show midpoints]; then you can toggle the lines' appearance on and off with that button and [Hide midpoints].

Pushing the blue radius control point to the left will make the blue circle and the a triangle smaller, and give a full triangle r0, r1 and r2 with non-zero area. The midpoint lines can be toggled as before. The red point controls the radius of the s circle and its base point s0 can be similarly dragged about to achieve a triangle of any desired shape. Many people seem to find it amusing to drag s0 or a0 rapidly round their circles when they are of different radii. (By doing so they are in fact exploring a torus in the Hopf fibration, but that is a more complicated story to tell in detail.)

The theory of the Discrete Fourier Transform of order 3, i.e., harmonic analysis in the cyclic group of order 3, a.k.a. the basic Geometric Fourier Transform that takes triangles to their equilateral and anti-equilateral harmonic components tells us that any triangle can be constructed in the way the mechanism here illustrates, provided its center of gravity is at the origin. That is by taking a standard equilateral triangle with positive orientation, rotating and dilating it (bigger or smaller), doing the same with the reverse orientation and averaging the two one can achieve any plane triangle centered on the origin.