Triangle

The triangle, or trig, is the simplest possible polygon, excluding the degenerate digon. Its highest symmetry version is called an equilateral triangle, to emphasize its three equal side lengths.

The equilateral triangle is one of the only three regular polygons that can tile the plane, the other two being the square and the hexagon. It's also the simplex of highest dimension that can tile its respective hyperplane in the triangular tiling.

Vertex coordinates
The vertices of a triangle of edge length 1 centered at the origin are
 * (±1/2, $\sqrt{3}$/6), (0, $\sqrt{3}$/3).

Other kinds of triangles
Beside the equilateral triangle, there are other kinds of triangles with non-equal edge lengths. Notably, these retain many of the properties of the highest-symmetry variant: any triangle is convex, has an inscribed and an exscribed circle, and tiles the plane. These properties don't generalize to any other polygon.

Isosceles triangle
An isosceles triangle has exactly two sides of equal length, and two of its three angles are identical. It has A1 symmetry.

Scalene triangle
A scalene triangle has three sides of different length, and three different angles. It has the trivial A0 symmetry.