Kepler Triangle

Author

Eric W. Weisstein
February 16, 2025

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

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

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

Definition

Reciprocal proportion triangle with proportion (ϕ^(1/2))

Figure

r=(GoldenRatio^(1/2));

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

Area

area=Area[tri]//RootReduce

N[area%,20]

ToRadicals[area]

(1/(2(Sqrt[GoldenRatio])))//RootReduce

Vertices

v=PolygonCoordinates[tri]

Graphics[MapIndexed[Text[#2[[1]],#]&,v]]

Angles

q=PolygonAngle[tri]//RootReduce

q//ToRadicals//Column

N[%,20]

ArcCos[(GoldenRatio^(-1/2))]//N

ArcCos[(GoldenRatio^(-1))]//N

Hypotenuse

((((GoldenRatio^(-1/2))))^2)+(1^2)//RootReduce

Figure

{1,(GoldenRatio^(1/2)),(GoldenRatio^(-1/2))}//N

Graphics[{FaceForm[LightYellow],EdgeForm[Black],tri,
Line[{{0,1},{1,1},{1,0}}.05],
Text[Subscript[α,3],v[[1]],-4{1,1}],
Text[Subscript[α,1],v[[3]],{5,-2}],
Text[Subscript[α,2],v[[2]],{-2,4}],
Text[Superscript[ϕ,Row[{-1,"/",2}]],Mean[v[[{1,2}]]],{2,0}],
Text[Superscript[ϕ,Row[{1,"/",2}]],Mean[v[[{2,3}]]],{-2,-2}],
Text[1,Mean[v[[{1,3}]]],{0,2}]
},
BaseStyle->Directive[FontFamily->"Times",18,Black],
ImageSize->500
]

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

RootReduce[1/r]