Ideal Triangle - Properties

Properties

In the standard hyperbolic plane (with Gaussian curvature -1 at every point):

• The interior angles of an ideal triangle are all zero.
• Any ideal triangle has area π.
• Any ideal triangle has infinite perimeter.
• The inscribed circle to an ideal triangle meets the triangle in three points of tangency, forming an equilateral triangle with side length

where is the golden ratio .

• The distance from any point in the triangle to the second-closest side of the triangle is less than or equal to d, with equality only for the three equilateral triangle vertices described above. The same inequality holds for hyperbolic triangles more generally; in a non-ideal triangle, the distance to the second-closest side is strictly less than d.

If the curvature is −K everywhere rather than −1, the areas above should be multiplied by 1/K and the lengths and distances should be multiplied by 1/√K.