# Geometry Through Art

### Inscribed Triangles

Just as there are color primaries, I believe that there are shape primaries as well -- at least the circle, the square, and the equilateral triangle (more later of the pentagon). All three are 'first' shapes of their kind: the circle is the first curved closed shape, the square the first polygon with an even number of equal-length sides, and the equilateral triangle the first polygon with an odd number of equal-length sides.

It is interesting to observe that given any square, the diagonals obtain the center, so that the centers coincide and a circle may easily be drawn either to inscribe the square or be inscribed by it. A much more elaborate procedure, however, is needed to do the same thing for triangles:

Given any circle, the inscribed triangle requires making first six points the chords of which delineate a hexagon. At least two triangles can be constructed using the diagonals of that hexagon:

Note that the chords do not intersect the center. The hexagon, having an even number of sides like the square, has diagonals that are diameters of the circle.

Continuing the inscribed triangle

List of Constructions