Spieker Center

Author

Eric W. Weisstein
January 3, 2007

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

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Incenter of the Medial Triangle

FullSimplify[SideLengths[Evaluate[TrilinearVertexMatrix[MedialTriangle]]],TriangleQ[]]

CircleFunction[Incircle]

..TBD...

Circumcenter of the Medial Triangle (But Want Incenter)

TrilinearVertexMatrix[MedialTriangle]

Factor[Numerator[Together[Expand[CircleEquation[{{0,(1/b),(1/c)},{(1/a),0,(1/c)},{(1/a),(1/b),0}}]]]]]

Coefficient[((a^4) (α^2)-(a^2) (b^2) (α^2)-(a^2) (c^2) (α^2)+2 a b (c^2) α β-(a^2) (b^2) (β^2)+(b^4) (β^2)-(b^2) (c^2) (β^2)+2 a (b^2) c α γ+2 (a^2) b c β γ-(a^2) (c^2) (γ^2)-(b^2) (c^2) (γ^2)+(c^4) (γ^2)),γ^2]

FullSimplify[LawOfCosines/@FullSimplify[CircleCenter[((a^4) (α^2)-(a^2) (b^2) (α^2)-(a^2) (c^2) (α^2)+2 a b (c^2) α β-(a^2) (b^2) (β^2)+(b^4) (β^2)-(b^2) (c^2) (β^2)+2 a (b^2) c α γ+2 (a^2) b c β γ-(a^2) (c^2) (γ^2)-(b^2) (c^2) (γ^2)+(c^4) (γ^2))]]]

DistanceTrilinears[CyclicTrilinears[2 a b c (Cos[A]+2 Cos[B] Cos[C])]][[1]]//Factor//FullSimplify

Nagel Line

CollinearQ[{NagelPoint,Incenter,Centroid,SpiekerCenter}]//Timing

Distances

SpCl

Factor[FullSimplify[Factor//@ToSidelengths[Distance[SpiekerCenter,ClawsonPoint]^2]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]],TriangleQ[]]]

ToAngles[%]

SpF

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,FeuerbachPoint]^2/.Cos[x_]:>TrigExpand[Cos[x]]],TriangleQ[]]]

ToAngles[%]

SpG

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,Centroid]^2],TriangleQ[]]]

ToAngles[%]

SpGe

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,GergonnePoint]^2],TriangleQ[]]]

ToAngles[%]

SpH

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,Orthocenter]^2],TriangleQ[]]]

ToAngles[%]

SpI

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,Incenter]^2],TriangleQ[]]]

ToAngles[%]

SpK

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,SymmedianPoint]^2],TriangleQ[]]]

ToAngles[%]

SpL

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,DeLongchampsPoint]^2],TriangleQ[]]]

ToAngles[%]

SpM

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,Mittenpunkt]^2],TriangleQ[]]]

ToAngles[%]

SpN

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,NinePointCenter]^2/.Cos[x_]:>TrigExpand[Cos[x]]],TriangleQ[]]]

ToAngles[%]

SpNa

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,NagelPoint]^2],TriangleQ[]]]

ToAngles[%]

SpO

Factor[FullSimplify[ToSidelengths[Distance[SpiekerCenter,Circumcenter]^2],TriangleQ[]]]

ToAngles[%]

SpSp

Distance[SpiekerCenter,SpiekerCenter]

Line through 3rd Brocard point and isotomic conjugate of the incenter

CollinearQ[{SpiekerCenter[tri],BrocardPoint[3][tri],IsotomicConjugatePoint[tri,Trilinears[Incenter]]}]//Timing

tri=Triangle[{{0.,0.},{.8,2.3},{3.,-.1}}];
v=Vertices[tri];
m=Vertices[MedialTriangle[tri]];
na=NagelPoint[tri];
q=Midpoint[{Coordinates@na,#}]&/@v;
pts={v[[1]],A,v[[2]],B,v[[3]],C,};
redpts={SpiekerCenter[tri],"\!\(\*
StyleBox[\"Sp\",\nFontSlant->\"Italic\"]\)",BrocardPoint[3][tri],Ω'',
IsotomicConjugatePoint[tri,Trilinears[Incenter]],"\!\(\*
StyleBox[\"I\",\nFontSlant->\"Italic\"]\)'"};

Show[Graphics[{
{
Green,
LongestLine[Coordinates/@{SpiekerCenter[tri],BrocardPoint[3][tri],IsotomicConjugatePoint[tri,Trilinears[Incenter]]}]
},
SpiekerCircle[tri],
PointLabels[tri,redpts,.03,.03]
}],
 AspectRatio->Automatic,PlotRange->All]

Cleavers

tri=Triangle[{{0.,0.},{.8,2.3},{3.,-.1}}];
v=Vertices[tri];
m=Vertices[MedialTriangle[tri]];
na=NagelPoint[tri];
q=Midpoint[{Coordinates@na,#}]&/@v;
pts={
Incenter[tri],"\!\(\*
StyleBox[\"I\",\nFontSlant->\"Italic\"]\)",
SpiekerCenter[tri],"\!\(\*
StyleBox[\"Sp\",\nFontSlant->\"Italic\"]\)",
v[[1]],A,v[[2]],B,v[[3]],C,
};
violet={m[[1]],A',m[[2]],B',m[[3]],C'};

Show[Graphics[{
tri,
{
Violet,
Triangle[m]
},
{
Red,
AngleBisectors[tri],
VertexArcs[tri,.3],
EqualAngleTicksAboutBisectors[tri,.3,.06,.1]
},
{
Blue,
Cleavers[tri],
PointLabels[tri,violet]
},
{
Dashing[{.02,.02}],
Incircle[tri],
SpiekerCircle[tri]
},
PointLabels[tri,pts]
}],
 AspectRatio->Automatic,PlotRange->All]

Structures Passing through It

pt=SpiekerCenter;

Circles

First/@Select[{#,IncidentCenters[{pt},#]}&/@Circles,Last[#]=!={}&]

Circle Centers

First/@Select[{#,IncidentCenters[{pt},CircleCenter[#]]}&/@Circles,Last[#]=!={}&]

Conics

First/@Select[{#,IncidentCenters[{pt},#]}&/@Conics,Last[#]=!={}&]

Cubics

First/@Select[{#,IncidentCenters[{pt},#]}&/@TriangleCubics,Last[#]=!={}&]

Lines

First/@Select[{#,IncidentCenters[{pt},#]}&/@Lines,Last[#]=!={}&]