Reciprocal Proportion Triangle

Author

Eric W. Weisstein
February 16, 2025

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polygons/ReciprocalProportionTriangle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ReciprocalProportionTriangle.html.

©2025 Wolfram Research, Inc. except for portions noted otherwise

Properties

tri=SSSTriangle[1/x,1,x];

Graphics[tri/.x->]

cond=FullSimplify[x>0&&And@@(#+#2>#3&@@@Partition[{1/x,1,x},3,1,1])]

NumberLinePlot[cond,x]

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

NumberLinePlot[x-1<(x^(-1))<x+1&&x>0,x]

#[{x,x-1<=(x^(-1))<=x+1&&x>0},x]&/@{MinValue,MaxValue}

N[%]

Select[
{E,Pi,GoldenRatio,Khinchin,GoldenAngle,Glaisher,plasticconstant=,supergoldenratio=,silverratio=(Sqrt[2])+1},
1/GoldenRatio<#<GoldenRatio&]

Angles

FullSimplify[PolygonAngle[tri],cond]

Area

FullSimplify[Area[tri],cond]//TraditionalForm

Perimeter

FullSimplify[Perimeter[tri],cond]

SideLengths

FullSimplify[EuclideanDistance@@@Subsets[PolygonCoordinates[tri],{2}],cond]