Musselman's Theorem

Author

Eric W. Weisstein
January 24, 2005

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

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

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

Graphic

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Red,ReflectedTriangle[t]},
{Green,Circumcircle[t]},
{Blue,Circumcircle[{Coordinates[Circumcenter[t]],Sequence@@#}]&/@Transpose[Vertices/@{t,ReflectedTriangle[t]}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ReflectedTriangle[t]],{"(SuperStar[A])","(SuperStar[B])","(SuperStar[C])"}}],
{
{Circumcenter[t],"O",Green},
{X[1157][t],"(Subscript[X, 1157])",Blue},
(*{NinePointCenter[t],"N",Green},*)
{Cartesian[IsogonalConjugate[Trilinears[NinePointCenter]][t]],"N'",Green}
}
]]
}],AspectRatio->Automatic,PlotRange->{{-2,1.5},{-1.5,2}}]

Derivation

Factor[Factor[Together//@ToSidelengths[InversePoint[IsogonalConjugate[NinePointCenter],Circle[Circumcircle]]/.f_Sec:>TrigExpand[f]]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]]

Symmetrize/@%

Symmetrize/@ToAngles[%]

IncidentCenters[KimberlingCenters,%]//Timing

CenterName[X[1157]]

CenterFunction[X[1157]]

TeXForm[((a^6)-3 (a^4) (b^2)+3 (a^2) (b^4)-(b^6)-3 (a^4) (c^2)-(a^2) (b^2) (c^2)+(b^4) (c^2)+3 (a^2) (c^4)+(b^2) (c^4)-(c^6)) Sec[B-C]]