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.