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Reciprocal Proportion Triangle

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The term "reciprocal proportion triangle" is used in this work to describe a triangle whose side lengths are in the proportion x:1:x^(-1). In order for such a triangle to exist, Tte triangle inequality requires

x-1<x^(-1)<x+1.
(1)

When combined with the requirement that side lengths be positive, the values of x for which a reciprocal proportion triangle exists are phi^(-1)<x<phi, i.e., 0.61803...<x<1.61803..., where phi is the golden ratio.

This means that reciprocal proportion triangles do not exist for common constants like e, pi, the golden ratio phi, and the Khinchin constant K. However, they do exist for the Glaisher-Kinkelin constant A, plastic constant P, supergolden ratio psi, and square root of the golden ratio sqrt(phi).

The following table summarizes some reciprocal proportion triangles.

xtriangle
supergolden ratio psisupergolden triangle
square root of golden ratio phi^(1/2)Kepler triangle
plastic constant Pplastic triangle

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

A=sqrt((-x^8+2x^6+x^4+2x^2-1)/(4x^2))
(2)

and angles

alpha_1=(3pi)/2+csc^(-1)((2x^3)/(x^4+x^2-1))
(3)
alpha_2=(3pi)/2+sin^(-1)((1-x^2+x^4)/(2x^2))
(4)
alpha_3=(3pi)/2+csc^(-1)((2x)/(1+x^2-x^4)).
(5)

Pegg (2016) terms such triangles "power triangles."

See also

Geometric Sequence, Plastic Constant, Supergolden Ratio, Triangle

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References

Pegg, E. Jr. "Wheels of Powered Triangles." 2016. https://demonstrations.wolfram.com/WheelsOfPoweredTriangles/.

Referenced on Wolfram|Alpha

Reciprocal Proportion Triangle

Cite this as:

Weisstein, Eric W. "Reciprocal Proportion Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ReciprocalProportionTriangle.html

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