Supergolden Triangle

Author

Eric W. Weisstein
February 16, 2025

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

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

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

Definition

Reciprocal proportion triangle with proportion ψ

Figure

psi=;

tri=SSSTriangle[1/psi,1,psi]//RootReduce;

Area

Area[tri]//RootReduce

Log[psi,%]//FullSimplify

N[%,20]

ToRadicals[]

Vertices

v=PolygonCoordinates[tri]

Angles

q=PolygonAngle[tri]//RootReduce

q//ToRadicals//Column

N[%,20]

Figure

Graphics[{FaceForm[LightYellow],EdgeForm[Black],tri,
Text[Subscript[α,1],v[[1]],{-6,-1.2}],
Text[Subscript[α,2],v[[3]],{5,-1.8}],
Text[Subscript[α,3],v[[2]],{.5,2}],
Text[1,Mean[v[[{1,2}]]],{-1,-3}],
Text[Superscript[ψ,-1],Mean[v[[{2,3}]]],{-2,-2}],
Text[ψ,Mean[v[[{1,3}]]],{0,2}]
},
BaseStyle->Directive[FontFamily->"Times",14,Black],
ImageSize->500
]

EuclideanDistance@@@Subsets[v,{2}]//RootReduce

RootReduce[1/psi]