Spieker Circle

Author

Eric W. Weisstein
January 12, 2005

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

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

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

Diagram

Medial Triangle

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],"\!\(\*
StyleBox[\"Sp\",\nFontSlant->\"Italic\"]\)"};
bluepts={
m[[1]],(Subscript[M, A]),m[[2]],(Subscript[M, B]),m[[3]],(Subscript[M, C])
};
violet={
q[[1]],(Subscript[Q, A]),q[[2]],(Subscript[Q, B]),q[[3]],(Subscript[Q, C])
};

Show[Graphics[{
tri,
{
Blue,
PointLabels[tri,bluepts],
Triangle[m]
},
{
Red,
SpiekerCircle[tri],
PointLabels[tri,redpts]
},
PointLabels[tri,pts],
EqualSideTicksAboutMidpoints[tri,.05,.015]
}],
 AspectRatio->Automatic,PlotRange->All]

Nagel Point Midpoints

t=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],"\!\(\*
StyleBox[\"I\",\nFontSlant->\"Italic\"]\)",
Centroid[tri],G,
v[[1]],A,v[[2]],B,v[[3]],C,
};
bluepts={na,"\!\(\*
StyleBox[\"Na\",\nFontSlant->\"Italic\"]\)"};
redpts={SpiekerCenter[tri],"\!\(\*
StyleBox[\"Sp\",\nFontSlant->\"Italic\"]\)"};
violet={q[[1]],(Subscript[Q, A]),q[[2]],(Subscript[Q, B]),q[[3]],(Subscript[Q, C])};

Show[Graphics[{
{
Green,
LongestLine[Coordinates/@Through[{NagelPoint,Incenter,Centroid,SpiekerCenter}[tri]]]
},
tri,
{
Blue,
Line[{Coordinates@na,#}]&/@v,
PointLabels[tri,bluepts],
Table[
EqualSideTicksAboutMidpoint[Line[{v[[i]],Coordinates@na}],i,.05],{i,3}]
},
{
Violet,
Triangle[q],
PointLabels[tri,violet]
},
{
Red,
SpiekerCircle[tri],
PointLabels[tri,redpts]
},
PointLabels[tri,pts]
}],
 AspectRatio->Automatic,PlotRange->All]

Theo's Triangular Table

Table Annotation

tri=Triangle[{{0.,0.},{0.,3.},{4.,0.}}];
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,};
violet={m[[1]],(Subscript[M, A]),m[[2]],(Subscript[M, B]),m[[3]],(Subscript[M, C])};
red={Incenter[tri],"\!\(\*
StyleBox[\"I\",\nFontSlant->\"Italic\"]\)"};
blue={SpiekerCenter[tri],"\!\(\*
StyleBox[\"Sp\",\nFontSlant->\"Italic\"]\)"};

Show[Graphics[{
tri,
{
Dashing[{.02,.02}],
Incircle[tri],
SpiekerCircle[tri]
},
{
Blue,
Cleavers[tri],
PointLabels[tri,blue]
},
{
Violet,
Triangle[m],
PerpSquare[Rest[Sides[Triangle@m]],.13,4],
PointLabels[tri,violet]
},
{
Red,
AngleBisectors[tri],
VertexArcs[tri,.3],
EqualAngleTicksAboutBisectors[tri,.3,.06,.1],
PerpSquare[Rest[Sides[tri]],.13,4],
PointLabels[tri,red]
},
PointLabels[tri,pts]
}],
 AspectRatio->Automatic,PlotRange->All]

Radius

r=Abs[FullSimplify[ToSidelengths[Distance[
Line@@(Midpoint/@{{{1,0,0},{0,1,0}},{{1,0,0},{0,0,1}}}),
Trilinears[SpiekerCenter]
]],TriangleQ[]]]

FullSimplify[ToSidelengths[r/Inradius],TriangleQ[]]

Circle Function

f=FullSimplify[Together//@DistanceTrilinears[CircleFunction[CircleEquation[Trilinears[SpiekerCenter],rr]]],TriangleQ[]]

Simplification

<<Algebra`SymmetricPolynomials`

SymmetricReduction[-(-3 (a^2)+5 (b^2)-6 b c+5 (c^2)+2 a (b+c)),{a,b,c}]

CircleFunction[SineTripleAngleCircle]

cos=CyclicTrilinears[Numerator[Together[#-DistanceTrilinears[#]&[Cos[A]Cos[B]Cos[C]]]]]

eqn=-(-3 (a^2)+5 (b^2)-6 b c+5 (c^2)+2 a (b+c));

Factor[GroebnerBasis[{eqn,Sequence@@cos},{a,b,c},MonomialOrder->EliminationOrder]]

sin=CyclicTrilinears[Numerator[Together[#-DistanceTrilinears[#]&[(Sin[A]Sin[B]Sin[C])^2]]]]

Factor[GroebnerBasis[{eqn,Sequence@@sin},{a,b,c},MonomialOrder->EliminationOrder]]

Rrs=Numerator[Together[{Circumradius[]^2-R^2,Inradius[]^2-r^2,Semiperimeter[]-s}]]

Factor[GroebnerBasis[{eqn,Sequence@@Rrs},{a,b,c},MonomialOrder->EliminationOrder]]

Together//@(4 b c-4 b s-4 c s+3 s^2/.s->Semiperimeter[]/.r->Inradius[])

Together//@(4(4 b c-4 b s-4 c s+3 s^2/.s->Semiperimeter[]/.r->Inradius[])-eqn)

eqn

Center Function

CircleFunction[SpiekerCircle]

InputForm[%]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[SpiekerCircle]]]//Timing

Center

Center[SpiekerCircle]

{CenterName[X[10]],CenterFunction[X[10]]}

FullSimplify[Together//@DistanceTrilinears[CircleCenter[CyclicTrilinears[-((-3 (a^2)+2 a b+5 (b^2)+2 a c-6 b c+5 (c^2))/(16 b c))]]]]

IncidentCenters[KimberlingCenters,CircleCenter[SpiekerCircle]]

Incident Points

(ctrs=IncidentCenters[KimberlingCenters,SpiekerCircle])//Timing

{#,CenterName[#]}&/@ctrs//TableForm