Reflection Circle

Author

Eric W. Weisstein
November 19, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Green,ReflectionTriangle[t]},
{Red,ReflectionCircle[t]},
{Blue,lines=Line/@Transpose[Vertices/@{t,ReflectionTriangle[t]}],
MapIndexed[EqualSideTicksAboutMidpoint[#1,#2[[1]],.02,.015]&,lines],
PerpSquare[#,.04,1]&/@Transpose[{lines,Sides[t]}]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ReflectionTriangle[t]],{"(SuperStar[A])","(SuperStar[B])","(SuperStar[C])"}}],
{
{Orthocenter[t],"H",{Blue,LabelOffset->{.06,-.05}}}
}
],LabelOffset->.04]
}],AspectRatio->Automatic]

Center

Center[ReflectionCircle][[1]]//TraditionalForm

IncidentCenters[KimberlingCenters,Center[ReflectionCircle]]//Timing

CenterName[X[195]]

Circle Function

CircleFunction[ReflectionCircle]//TraditionalForm

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

Radius

CircleRadius[ReflectionCircle]//TraditionalForm

Refine[Sqrt[ToAngles[Factor[ToSidelengths[(1/(S^3 (Distance[Circumcenter,Orthocenter]^2 - 4 Circumradius^2)))Sqrt[Times@@((S^2 (Distance[Circumcenter,Orthocenter]^2 + 2 #) + 2 SA SB SC)&/@{SA,SB,SC})]]^2]]],TriangleQ[]&&Area>0]//TraditionalForm

Powers

Factor[Simplify[Factor[ToSidelengths[CirclePowers[ReflectionCircle]]],TriangleQ[]]]

TraditionalForm[%]

Factor[ToSidelengths[(8 S^4 - b^2 c^2 (3 S^2 + SB SC)) / (2 (OH^2 - 4 Circumradius^2) S^2)/.OH->((Sqrt[(a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-(a^4) (c^2)+3 (a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)])/(4 Area))]]//TraditionalForm

Incident Centers

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

pts=Flatten[{Anticomplement/@{X[110],X[930]},Reflection[X[146],X[4]],Reflection[X[3],X[1263]]}];

Together//@ToSidelengths[(TrilinearEquation[ReflectionCircle]/.Thread[{α,β,γ}->#])&/@pts/.f:(_Cos|_Csc):>TrigExpand[f]]

Derivation

Factor[TrilinearEquation@@ToSidelengths[TrilinearVertexMatrix[ReflectionTriangle]]]

eqn=((a^10) (α^2)-4 (a^8) (b^2) (α^2)+6 (a^6) (b^4) (α^2)-4 (a^4) (b^6) (α^2)+(a^2) (b^8) (α^2)-4 (a^8) (c^2) (α^2)+5 (a^6) (b^2) (c^2) (α^2)+(a^4) (b^4) (c^2) (α^2)-2 (a^2) (b^6) (c^2) (α^2)+6 (a^6) (c^4) (α^2)+(a^4) (b^2) (c^4) (α^2)+2 (a^2) (b^4) (c^4) (α^2)-4 (a^4) (c^6) (α^2)-2 (a^2) (b^2) (c^6) (α^2)+(a^2) (c^8) (α^2)+2 (a^9) b α β-8 (a^7) (b^3) α β+12 (a^5) (b^5) α β-8 (a^3) (b^7) α β+2 a (b^9) α β-7 (a^7) b (c^2) α β+7 (a^5) (b^3) (c^2) α β+7 (a^3) (b^5) (c^2) α β-7 a (b^7) (c^2) α β+9 (a^5) b (c^4) α β+3 (a^3) (b^3) (c^4) α β+9 a (b^5) (c^4) α β-5 (a^3) b (c^6) α β-5 a (b^3) (c^6) α β+a b (c^8) α β+(a^8) (b^2) (β^2)-4 (a^6) (b^4) (β^2)+6 (a^4) (b^6) (β^2)-4 (a^2) (b^8) (β^2)+(b^10) (β^2)-2 (a^6) (b^2) (c^2) (β^2)+(a^4) (b^4) (c^2) (β^2)+5 (a^2) (b^6) (c^2) (β^2)-4 (b^8) (c^2) (β^2)+2 (a^4) (b^2) (c^4) (β^2)+(a^2) (b^4) (c^4) (β^2)+6 (b^6) (c^4) (β^2)-2 (a^2) (b^2) (c^6) (β^2)-4 (b^4) (c^6) (β^2)+(b^2) (c^8) (β^2)+2 (a^9) c α γ-7 (a^7) (b^2) c α γ+9 (a^5) (b^4) c α γ-5 (a^3) (b^6) c α γ+a (b^8) c α γ-8 (a^7) (c^3) α γ+7 (a^5) (b^2) (c^3) α γ+3 (a^3) (b^4) (c^3) α γ-5 a (b^6) (c^3) α γ+12 (a^5) (c^5) α γ+7 (a^3) (b^2) (c^5) α γ+9 a (b^4) (c^5) α γ-8 (a^3) (c^7) α γ-7 a (b^2) (c^7) α γ+2 a (c^9) α γ+(a^8) b c β γ-5 (a^6) (b^3) c β γ+9 (a^4) (b^5) c β γ-7 (a^2) (b^7) c β γ+2 (b^9) c β γ-5 (a^6) b (c^3) β γ+3 (a^4) (b^3) (c^3) β γ+7 (a^2) (b^5) (c^3) β γ-8 (b^7) (c^3) β γ+9 (a^4) b (c^5) β γ+7 (a^2) (b^3) (c^5) β γ+12 (b^5) (c^5) β γ-7 (a^2) b (c^7) β γ-8 (b^3) (c^7) β γ+2 b (c^9) β γ+(a^8) (c^2) (γ^2)-2 (a^6) (b^2) (c^2) (γ^2)+2 (a^4) (b^4) (c^2) (γ^2)-2 (a^2) (b^6) (c^2) (γ^2)+(b^8) (c^2) (γ^2)-4 (a^6) (c^4) (γ^2)+(a^4) (b^2) (c^4) (γ^2)+(a^2) (b^4) (c^4) (γ^2)-4 (b^6) (c^4) (γ^2)+6 (a^4) (c^6) (γ^2)+5 (a^2) (b^2) (c^6) (γ^2)+6 (b^4) (c^6) (γ^2)-4 (a^2) (c^8) (γ^2)-4 (b^2) (c^8) (γ^2)+(c^10) (γ^2));

circ=Circumcircle[eqn];

ctr=Factor[Center[circ][[1]]]

-Symmetrize[ctr]//InputForm

InputForm[ctr]

IncidentCenters[KimberlingCenters,CyclicTrilinears[ctr]]//Timing

CenterName[X[195]]

CenterFunction[X[195]]

Factor[CircleFunction[eqn][[1]]]

InputForm[%]

r2=Factor[Together//@(CircleRadius[circ]^2)]

FullSimplify[Sqrt[r2],TriangleQ[]]

Map[Factor,Refine[ToAngles[%189],TriangleQ[]&&Area>0],{3}]

InputForm[%203]