Stammler Circles

Author

Eric W. Weisstein
January 24, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/StammlerCircles.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Figure

t=Triangle[{{0,0},{1,0},{.15,.7}}];
Show[Graphics[{
{Purple,Sides[t,2]},
t,
{Red,circs=StammlerCircles[t,{1,2,3}]},
{Blue,Circumcircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"},{LabelOffset->-.1{1,1},LabelOffset->.1{1,-1},{}}}],
Append[#,Red]&/@Transpose[{Center/@circs,{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"}}]
]]
}],AspectRatio->Automatic,PlotRange->{{-1.5,2.5},{-1.9,2.2}}]

Chord Lengths

t=Triangle[{{0,0},{1,0},{.15,.7}}];
SideLengths[Evaluate[t]]
circs=StammlerCircles[t,{1,2,3}];Table[Norm[Subtract@@({x,y}/.Solve[#.#&[First[#]-{x,y}]==Last[#]^2&[circs[[i]]]&&(y-#[[1,2]])(#[[1,1]]-#[[2,1]])==(x-#[[1,1]])(#[[1,2]]-#[[2,2]])&[Sides[t][[s,1]]],{x,y}])],{i,3},{s,3}]

t=Triangle[{{1,0},{-.15,.7},{0,0}}];
SideLengths[Evaluate[t]]
circs=StammlerCircles[t,{2,3,1}];
Table[Norm[Subtract@@({x,y}/.Solve[#.#&[First[#]-{x,y}]==Last[#]^2&[circs[[i]]]&&(y-#[[1,2]])(#[[1,1]]-#[[2,1]])==(x-#[[1,1]])(#[[1,2]]-#[[2,2]])&[Sides[t][[s,1]]],{x,y}])],{i,3},{s,3}]

Relationships Between Radii

{RA,RB,RC}=CircleRadius/@Circles[StammlerCircles];

((Subscript[R, A])^2)+((Subscript[R, B])^2)+((Subscript[R, C])^2)==9 (R^2)

FullSimplify[TrigExpand[(RA^2)+(RB^2)+(RC^2)]/.C->π-A-B]

(1/((Subscript[R, A])^2))+(1/((Subscript[R, B])^2))+(1/((Subscript[R, C])^2))==((3 ((a^2)+(b^2)+(c^2)))/((a^2)(b^2)+(c^2) (a^2)+(b^2) (c^2)))

FullSimplify[Factor[(1/(RA^2))+(1/(RB^2))+(1/(RC^2))]/.C->π-A-B]

(Subscript[R, A]) (Subscript[R, B]) (Subscript[R, C])==R (Sqrt[(a^2) (b^2)+(c^2) (b^2)+(a^2) (c^2)])

FullSimplify[ToSidelengths[%],TriangleQ[]]

FullSimplify[Factor[(RA^2)(RB^2)(RC^2)]/.C->π-A-B]

FullSimplify[ToSidelengths[%],TriangleQ[]]

Derivation

t=Triangle[{{1,0},{-.15,.7},{0,0}}];

{a,b,c}=EuclideanDistance@@@Partition[t[[1]],2,1,1]

R=Circumsphere[t][[2]]

x/.{ToRules[NRoots[(x^3)-(R^2)(9(x^2)-3((a^2)+(b^2)+(c^2))x+((a^2)(b^2)+(b^2)(c^2)+(c^2)(a^2))),x]]}

Sqrt[%]

StammlerCircles[t]

ToSidelengths[Sqrt[{(1+8 Cos[(B-C)/3] Cos[(B+2 C)/3] Cos[(2 B+C)/3]) Circumradius^2,
(1+8 Cos[(C-A)/3] Cos[(C+2 A)/3] Cos[(2 C+A)/3]) Circumradius^2,
(1+8 Cos[(A-B)/3] Cos[(A+2 B)/3] Cos[(2 A+B)/3]) Circumradius^2}]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]