Mandart Circle

Author

Eric W. Weisstein
December 17, 2004

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

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

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

Circle Function

CircleFunction[MandartCircle]

IncidentCenters[{X[221]},CyclicTrilinears[CircleFunction[MandartCircle]]]//Timing

CenterName[X[221]]

Circle Center

Symmetrize[Center[MandartCircle][[1]]]

IncidentCenters[KimberlingCenters,CircleCenter[MandartCircle]]//Timing

{CenterName[X[1158]],Symmetrize[CenterFunction[X[1158]]]}

Other Constructions

Cartesian[PedalTriangle[X[84]][t]]

Cartesian[CevianTriangle[X[189]][t]]

t=Triangle[{{0,0},{5/10,35/100},{0,1}}]//N[#,10]&;
g1=Show[Graphics[{
t,
{Green,pt=Cartesian[PedalTriangle[X[84]][t]]},
{Red,Circumcircle[pt]},
{Orange,ExtouchTriangle[t]},
{Purple,Cartesian[#[t]]&/@Cevians[X[189]]/.Line[x_]:>LongestLine[Append[x,X[189][t][[1]]]]},
{Blue,Cartesian[#[t]]&/@PedalLines[X[84]]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Green]&/@Transpose[{VertexPoints[pt],{"(Subscript[P, A])","(Subscript[P, B])","(Subscript[P, C])"}}],
Append[#,Orange]&/@Transpose[{VertexPoints[Cartesian[PedalTriangle[X[40]][t]]],{"A'","B'","C'"}}],
{
{X[84][t],"V",Blue},
{X[189][t],"(Subscript[X, 189])",Purple}
}
],LabelOffset->.04,PointSize[.02]]
}],AspectRatio->Automatic,PlotRange->{{-.1,1.85},{-.1,1.1}}]

Incident Centers

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

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

Check

Short/@(FullSimplify[ToSidelengths[TrigExpand[TrilinearEquation[MandartCircle]/.Thread[{α,β,γ}->Trilinears[#]]]],
TriangleQ[]]&/@ctrs)//Timing

Figure

t=Triangle[{{0,0},{1,0},{.25,.5}}];
Show[Graphics[{
{Purple,Sides[t,2]},
t,
{Red,MandartCircle[t]},
{Green,Excircles[t]},
{Blue,ExtouchTriangle[t]},
{Yellow,Incircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[ExtouchTriangle[t]],{"(Subscript[T, A])","(Subscript[T, B])","(Subscript[T, C])"}}],
{
{X[1158][t],"(Subscript[X, 1158])",Red},
{X[11][t],"F",Red},
{X[1364][t],"(Subscript[X, 1364])",Red}
}
],PointSize[.025],LabelOffset->.05]
}],AspectRatio->Automatic,PlotRange->{{-.1,1.2},{-.2,.8}}]

Derivation from Scratch

eqn=Factor[Simplify[((128 (((a b c))^2) (((a α+b β+c γ))^2)Circumradius[])/(((Semiperimeter[])^2)Times@@CyclicTrilinears[(((a+b-c))^2)]))Together//@CircleEquation[TrilinearVertexMatrix[ExtouchTriangle]],TriangleQ[]]]

Circle Function

CircleFunction[ExtouchTriangleCircumcircle]

InputForm[%]

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

{CenterName[X[221]],CenterFunction[X[221]]}

Symmetrize[Together//@DistanceTrilinears[HalfAngleFormulas[CenterFunction[X[221]]]]]

Symmetrize[CircleFunction[ExtouchTriangleCircumcircle]]

Incident Centers

IncidentCenters[KimberlingCenters,ExtouchTriangleCircumcircle]//Timing

Center

center=(1/2)Numerator[DistanceTrilinears[CircleCenter[eqn][[1]]]]//Factor

CenterName[X[1158]]

CenterFunction[X[1158]]//Factor

Factor[2(a b c)Times@@CyclicTrilinears[a+b-c]DistanceTrilinears[CircleCenter[ExtouchTriangleCircumcircle][[1]]]]

Radius

Simplify[Together//@ToSidelengths[CircleRadius[ExtouchTriangleCircumcircle]],TriangleQ[]]

FullSimplify[%,TriangleQ[]]

expr=((Sqrt[-(((((((a^2)-(b^2)))^2)-2 (((a-b))^2) (c^2)+(c^4)) (((((a^2)-(b^2)))^2)+4 a (b^2) c-2 ((a^2)+(b^2)) (c^2)+(c^4)) ((a^4)-2 (a^2) (((b-c))^2)+(((b-c))^2) (((b+c))^2)))/((((a-b-c))^3) (a+b+c)))])/(2 ((((a+b-c) (a-b+c)))^(3/2))));

FullSimplify[expr,TriangleQ[]]

FullSimplify[expr,b+c>a]

Times@@Factor[(((((a^2)-(b^2)))^2)-2 (((a-b))^2) (c^2)+(c^4)) ((a^4)-2 (a^2) (((b-c))^2)+((((b^2)-(c^2)))^2)) ((a^4)+4 a (b^2) c+((((b^2)-(c^2)))^2)-2 (a^2) ((b^2)+(c^2)))]

(1/(2 (Sqrt[a+b+c])((((a+b-c) (a-b+c) (-a+b+c)))^(3/2))))

FullSimplify[((Sqrt[(1/(a+b+c))])/(2 ((((a+b-c) (a-b+c) (-a+b+c)))^(3/2)))),TriangleQ[]]

ToAngles[%28]

((a^4)-2 (a^2) (b^2)+(b^4)+4 (a^2) b c-2 (a^2) (c^2)-2 (b^2) (c^2)+(c^4)) ((a^4)-2 (a^2) (b^2)+(b^4)+4 a (b^2) c-2 (a^2) (c^2)-2 (b^2) (c^2)+(c^4)) ((a^4)-2 (a^2) (b^2)+(b^4)-2 (a^2) (c^2)+4 a b (c^2)-2 (b^2) (c^2)+(c^4))/(2 ((((a+b-c) (a-b+c) (-a+b+c)))^(3/2)) (Sqrt[a+b+c]))//InputForm

Times@@CyclicTrilinears[((a^4)-2 (a^2) (b^2)+(b^4)+4 (a^2) b c-2 (a^2) (c^2)-2 (b^2) (c^2)+(c^4)) ]

Derivation Given Center and One Point

CenterName[X[1158]]

CenterFunction[X[1158]]

Symmetrize[CenterFunction[X[1158]]]//InputForm

circ=Circumcircle[ExtouchTriangle];//Timing

(Together//@ToSidelengths[(Distance[X[11],X[1158]]/.Cos[x_]:>TrigExpand[Cos[x]])])/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x]Sqrt[y],TriangleQ[]]

(√Factor[(((a^12)-6 (a^10) (b^2)+15 (a^8) (b^4)-20 (a^6) (b^6)+15 (a^4) (b^8)-6 (a^2) (b^10)+(b^12)+4 (a^10) b c+4 (a^9) (b^2) c-16 (a^8) (b^3) c-16 (a^7) (b^4) c+24 (a^6) (b^5) c+24 (a^5) (b^6) c-16 (a^4) (b^7) c-16 (a^3) (b^8) c+4 (a^2) (b^9) c+4 a (b^10) c-6 (a^10) (c^2)+4 (a^9) b (c^2)+18 (a^8) (b^2) (c^2)-12 (a^6) (b^4) (c^2)-8 (a^5) (b^5) (c^2)-12 (a^4) (b^6) (c^2)+18 (a^2) (b^8) (c^2)+4 a (b^9) (c^2)-6 (b^10) (c^2)-16 (a^8) b (c^3)+32 (a^6) (b^3) (c^3)-16 (a^5) (b^4) (c^3)-16 (a^4) (b^5) (c^3)+32 (a^3) (b^6) (c^3)-16 a (b^8) (c^3)+15 (a^8) (c^4)-16 (a^7) b (c^4)-12 (a^6) (b^2) (c^4)-16 (a^5) (b^3) (c^4)+58 (a^4) (b^4) (c^4)-16 (a^3) (b^5) (c^4)-12 (a^2) (b^6) (c^4)-16 a (b^7) (c^4)+15 (b^8) (c^4)+24 (a^6) b (c^5)-8 (a^5) (b^2) (c^5)-16 (a^4) (b^3) (c^5)-16 (a^3) (b^4) (c^5)-8 (a^2) (b^5) (c^5)+24 a (b^6) (c^5)-20 (a^6) (c^6)+24 (a^5) b (c^6)-12 (a^4) (b^2) (c^6)+32 (a^3) (b^3) (c^6)-12 (a^2) (b^4) (c^6)+24 a (b^5) (c^6)-20 (b^6) (c^6)-16 (a^4) b (c^7)-16 a (b^4) (c^7)+15 (a^4) (c^8)-16 (a^3) b (c^8)+18 (a^2) (b^2) (c^8)-16 a (b^3) (c^8)+15 (b^4) (c^8)+4 (a^2) b (c^9)+4 a (b^2) (c^9)-6 (a^2) (c^10)+4 a b (c^10)-6 (b^2) (c^10)+(c^12))/((a^6)-2 (a^5) b-(a^4) (b^2)+4 (a^3) (b^3)-(a^2) (b^4)-2 a (b^5)+(b^6)-2 (a^5) c+6 (a^4) b c-4 (a^3) (b^2) c-4 (a^2) (b^3) c+6 a (b^4) c-2 (b^5) c-(a^4) (c^2)-4 (a^3) b (c^2)+10 (a^2) (b^2) (c^2)-4 a (b^3) (c^2)-(b^4) (c^2)+4 (a^3) (c^3)-4 (a^2) b (c^3)-4 a (b^2) (c^3)+4 (b^3) (c^3)-(a^2) (c^4)+6 a b (c^4)-(b^2) (c^4)-2 a (c^5)-2 b (c^5)+(c^6)))])/(2 (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))

(1/2) √(((a^4)-2 (a^2) (b^2)+(b^4)+4 (a^2) b c-2 (a^2) (c^2)-2 (b^2) (c^2)+(c^4)) ((a^4)-2 (a^2) (b^2)+(b^4)+4 a (b^2) c-2 (a^2) (c^2)-2 (b^2) (c^2)+(c^4)) ((a^4)-2 (a^2) (b^2)+(b^4)-2 (a^2) (c^2)+4 a b (c^2)-2 (b^2) (c^2)+(c^4))/((((a-b-c))^3) (((a+b-c))^3) (((a-b+c))^3) (a+b+c)))//InputForm

((Semiperimeter[])/(2a b c))(Sqrt[Factor[Times@@Together[(4Circumradius[]^2-#)&/@{b c,c a,a b}]]])

Map[Factor,Simplify[%35,TriangleQ[]],2]

off by 2

r=Map[Factor,FullSimplify[Together//@(((Semiperimeter[])/(a b c))(Sqrt[(4(R^2)-b c)(4(R^2)-c a)(4(R^2)-a b)])/.R->Circumradius[]),TriangleQ[]],2]

CircleFunction[circ_Circle]:=CircleFunction[TrilinearEquation[circ]]

circ=Circle[X[1158],r];

Factor[Together//@ToSidelengths[CircleFunction[circ]]]

Factor[%/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]

Symmetrize/@%52

InputForm[%52[[1]]]

Symmetrize[ToSidelengths[HalfAngleFormulas[CenterFunction[X[221]]]]]