Ideal triangle

Properties

Ideal triangles have the following properties:

• All ideal triangles are congruent to each other.
• The interior angles of an ideal triangle are all zero.
• An ideal triangle has infinite perimeter.
• An ideal triangle is the largest possible triangle in hyperbolic geometry.

In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties:

• Any ideal triangle has area π.

Distances in an ideal triangle

• The inscribed circle to an ideal triangle has radius

${\displaystyle r=\ln \sqrt{3}=\frac{1}{2}\ln 3=\mathrm{artanh}\frac{1}{2}=2\mathrm{artanh}(2-\sqrt{3})=}$ ${\displaystyle =\mathrm{arsinh}\frac{1}{3}\sqrt{3}=\mathrm{arcosh}\frac{2}{3}\sqrt{3}\approx 0.549}$ .

The distance from any point in the triangle to the closest side of the triangle is less than or equal to the radius r above, with equality only for the center of the inscribed circle.

• The inscribed circle meets the triangle in three points of tangency, forming an equilateral contact triangle with side length ${\displaystyle d=\ln \left(\frac{\sqrt{5}+1}{\sqrt{5}-1}\right)=2\ln \phi \approx 0.962}$ where ${\displaystyle \phi =\frac{1+\sqrt{5}}{2}}$ is the golden ratio.

A circle with radius d around a point inside the triangle will meet or intersect at least two sides of the triangle.

• The distance from any point on a side of the triangle to another side of the triangle is equal or less than ${\displaystyle a=\ln \left(1+\sqrt{2}\right)\approx 0.881}$, with equality only for the points of tangency described above.

a is also the altitude of the Schweikart triangle.