Yiu Circles

Author

Eric W. Weisstein
February 6, 2005

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

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

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

Figure

pts=Reflection[Last[#],TrilinearLine[Line@@Take[#,2]]]&/@Partition[IdentityMatrix[3],3,1,1]

t=Triangle[Reverse/@{{0,0},{.16,.5},{1,0}}];

lines=Cartesian[#[t]]&/@(Line@@@Transpose[{pts,RotateLeft[IdentityMatrix[3],2]}]);

Show[Graphics[{
t,
{Purple,YiuCircles[t]},
Transpose[{{Blue,Red,Green},lines}],
Transpose[{{Red,Green,Blue},
MapIndexed[EqualSideTicksAboutMidpoint[#1,#2[[1]],.04,.03]&,RotateLeft[lines,1]]}],
Transpose[{{Red,Green,Blue},PerpSquare[#,.08,1]&/@Transpose[{Sides[t],RotateLeft[lines,1]}]}],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"},{{Red,LabelOffset->{0,-.15}},Green,{Blue,LabelOffset->{.12,-.02}}}}],
Transpose[{Cartesian[#[t]]&/@pts,{"C'","A'","B'"},{Blue,Red,Green}}],
Transpose[{Center/@YiuCircles[t],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"},{Red,Green,{Blue,LabelOffset->{-.12,-.02}}}}]]]
}],AspectRatio->Automatic,PlotRange->All,ImageSize->450]

Circles

Circles[YiuCircles]//TraditionalForm

Radii

FullSimplify[ToAngles[CircleRadius[Circles[YiuCircles][[1]]]],TriangleQ[]]

(√Factor[(a^6)-3 (a^4) ((b^2)+(c^2))-((((b^2)-(c^2)))^2) ((b^2)+(c^2))+3 (a^2) ((b^4)+(b^2) (c^2)+(c^4))] Csc[A])/(2 b c Abs[1+2 Cos[2 A]])(((1/2)b c /Area)/(Csc[A]))

InputForm[%18]

Refine[Sqrt[ToAngles[Factor[Together//@ToSidelengths[((b^4)(c^4)((S^2)(((Distance[Circumcenter,Orthocenter])^2)+2SA)+2SA SB SC))/((S^2)((((b^2)(c^2)-4(SA^2)))^2))]]](((((((-(a^2)+(b^2)-b c+(c^2)) (-(a^2)+(b^2)+b c+(c^2)))/((b^2) (c^2)))/(1+2Cos[2A]))))^2)],TriangleQ[]&&Area>0]

Circle Powers

Factor[Factor[ToSidelengths[CirclePowers[Circles[YiuCircles][[1]]]/.f_Cos:>TrigExpand[f]]]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]/. 1/Abs[x_]^2:>1/x^2 /.Abs[x_]^2:>x^2]

Simplify[-ToAngles[%]]

Simplify[-ToAngles[Factor[ToSidelengths[{0, 2 c^2 (S^2 + SC SA), 2 b^2 (S^2 + SA SB)} / (b^2 c^2 - 4 SA^2)]]]]

Radical Center

t=Triangle[{{0,0},{1,0},{16/100,5/10}}]//N[#,50]&;
Show[Graphics[{
t,
{Blue,YiuCircles[t]},
{Red,lines=RadicalLines[YiuCircles[t],5]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{Center/@YiuCircles[t],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"}}],
{
{Intersections@@Take[lines,2],"X",Red}
}
],PointSize[.022]]
}],AspectRatio->Automatic,PlotRange->{{-2.5,1.5},{-2,2.5}}]

eqns=Factor[Together//@ToSidelengths[TrilinearEquation/@Circles[YiuCircles]/.Cos[x_]:>TrigExpand[Cos[x]]]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]/.Sqrt[x_]:>Simplify[Sqrt[x],TriangleQ[]]];

eqns2=Factor[eqns/.Abs[x_]^2:>x^2]

lmn=Factor[CircleFunction/@eqns2]

ctr=Factor[RadicalCenter[lmn]]

ctr={-(((a^4)-2 (a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)) (-(a^4)+(a^2) (b^2)+2 (a^2) (c^2)+(b^2) (c^2)-(c^4)) (-(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)))/(b c ((a^2)-a b+(b^2)-(c^2)) ((a^2)+a b+(b^2)-(c^2)) (-(a^2)+(b^2)-a c-(c^2)) (-(a^2)+(b^2)+a c-(c^2)) (-(a^2)+(b^2)-b c+(c^2)) (-(a^2)+(b^2)+b c+(c^2))),-(((a^4)-2 (a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)) ((a^2) (b^2)-(b^4)+(a^2) (c^2)+2 (b^2) (c^2)-(c^4)) ((a^6)-3 (a^4) (b^2)+3 (a^2) (b^4)-(b^6)-(a^4) (c^2)+(a^2) (b^2) (c^2)+3 (b^4) (c^2)-(a^2) (c^4)-3 (b^2) (c^4)+(c^6)))/(a c ((a^2)-a b+(b^2)-(c^2)) ((a^2)+a b+(b^2)-(c^2)) ((a^2)-(b^2)-b c-(c^2)) ((a^2)-(b^2)+b c-(c^2)) ((a^2)-(b^2)-a c+(c^2)) ((a^2)-(b^2)+a c+(c^2))),-(((a^2) (b^2)-(b^4)+(a^2) (c^2)+2 (b^2) (c^2)-(c^4)) ((a^4)-(a^2) (b^2)-2 (a^2) (c^2)-(b^2) (c^2)+(c^4)) ((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-3 (a^4) (c^2)+(a^2) (b^2) (c^2)-3 (b^4) (c^2)+3 (a^2) (c^4)+3 (b^2) (c^4)-(c^6)))/(a b ((a^2)-a b+(b^2)-(c^2)) ((a^2)+a b+(b^2)-(c^2)) ((a^2)-(b^2)-b c-(c^2)) ((a^2)-(b^2)+b c-(c^2)) ((a^2)-(b^2)-a c+(c^2)) ((a^2)-(b^2)+a c+(c^2)))};

Symmetrize[(((a^4)-2 (a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)) (-(a^4)+(a^2) (b^2)+2 (a^2) (c^2)+(b^2) (c^2)-(c^4)) (-(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)))/(b c ((a^2)-a b+(b^2)-(c^2)) ((a^2)+a b+(b^2)-(c^2)) (-(a^2)+(b^2)-a c-(c^2)) (-(a^2)+(b^2)+a c-(c^2)) (-(a^2)+(b^2)-b c+(c^2)) (-(a^2)+(b^2)+b c+(c^2)))]

Symmetrize[ToAngles[%]]

IncidentCenters[KimberlingCenters,ctr]//Timing

CenterName[X[1157]]

Radical Circle is the Radical Center

circs=Circles[YiuCircles]

t=Triangle[{{0,0},{1,0},{16/100,5/10}}]//N[#,50]&;

Sqrt[Distance[Center[YiuCirclesRadicalCircle],Center[#]]^2-CircleRadius[#]^2&[#]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]]&/@circs

r=Sqrt[Distance[Center[YiuCirclesRadicalCircle],Center[#]]^2-CircleRadius[#]^2&[circs[[1]]]]

r2=Together//@ToSidelengths[r/.{Cos[x_]:>TrigExpand[Cos[x]],Sec[x_]:>TrigExpand[Sec[x]]}]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]/.Sqrt[x_]:>Simplify[Sqrt[x],TriangleQ[]]/.Abs[x_]^2:>x^2

Derivation

pts=Reflection[Last[#],TrilinearLine[Line@@Take[#,2]]]&/@Partition[IdentityMatrix[3],3,1,1]

Factor[Together//@ToSidelengths[TrilinearEquation[pts[[1]],{1,0,0},pts[[3]]]]]

eqn=(-(a^3) (b^2) c α β+a (b^4) c α β+2 a (b^2) (c^3) α β-(a^4) b c (β^2)+(a^2) (b^3) c (β^2)+2 (a^2) b (c^3) (β^2)+(b^3) (c^3) (β^2)-b (c^5) (β^2)-(a^3) b (c^2) α γ+2 a (b^3) (c^2) α γ+a b (c^4) α γ+(a^6) β γ-3 (a^4) (b^2) β γ+3 (a^2) (b^4) β γ-(b^6) β γ-3 (a^4) (c^2) β γ+3 (a^2) (b^2) (c^2) β γ+(b^4) (c^2) β γ+3 (a^2) (c^4) β γ+(b^2) (c^4) β γ-(c^6) β γ-(a^4) b c (γ^2)+2 (a^2) (b^3) c (γ^2)-(b^5) c (γ^2)+(a^2) b (c^3) (γ^2)+(b^3) (c^3) (γ^2));

Factor[Center[eqn]]

Factor[(1/(a (b^2)(c^2)))Simplify[ToAngles[%]]]

InputForm[4%190]

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

Map[Factor,%,3]

ToAngles[%]

Simplify[%,TriangleQ[]]

Map[Factor,%,3]

InputForm[%]

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
PointSize[.02],Cartesian[#[t]]&/@pts
}],AspectRatio->Automatic]