Reciprocal Proportion Triangle

The term "reciprocal proportion triangle" is used in this work to describe a triangle whose side lengths are in the proportion . In order for such a triangle to exist, the triangle inequality requires

(1)

When combined with the requirement that side lengths be positive, the values of for which a reciprocal proportion triangle exists are , i.e., , where is the golden ratio.

This means that reciprocal proportion triangles do not exist for common constants like , , the golden ratio , and the Khinchin constant . However, they do exist for the Glaisher-Kinkelin constant , plastic constant , supergolden ratio , and square root of the golden ratio .

The following table summarizes some reciprocal proportion triangles.

	triangle
supergolden ratio	supergolden triangle
square root of golden ratio	Kepler triangle
plastic constant	plastic triangle

When a reciprocal proportion triangle exists with ratio , it has area

(2)

and angles

			(3)
			(4)
			(5)

Pegg (2016) terms triangles whose side lengths are powers of a common base "power triangles." For example, the power triangle with base and exponents 0, 1, and 2 has side lengths , , and . A number of power triangles have exceptional additional dissections into similar triangles (E. Pegg, Jr., pers. comm., Sep. 28, 2026). Iterating such a dissection gives a multiscale substitution tiling rule because the child triangles can have different scale factors.