Outer Vecten Circle

Author

Eric W. Weisstein
January 19, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{3/10,8/10}}];
Show[Graphics[{
{Blue,ErectPolygons[Vertices@t,4,-1]},
t,
{Green,OuterVectenTriangle[t]},
{Red,OuterVectenCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[OuterVectenTriangle[t]],{"(Subscript[O, C])","(Subscript[O, A])","(Subscript[O, B])"}}],
{
{Center[OuterVectenCircle[t]],"(Subscript[X, 641])",Red}
}
],PointSize[.025]]
},AspectRatio->Automatic]
]

Center

Center[OuterVectenCircle]//TraditionalForm

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

CenterName/@{X[485],X[641]}

Circle Function

CircleFunction[OuterVectenCircle]//TraditionalForm

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

Radius

CircleRadius[OuterVectenCircle]//TraditionalForm

Incident Centers

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

Derivation

v=FullSimplify[4ToSidelengths[TrilinearVertexMatrix[OuterVectenTriangle]],TriangleQ[]]/.(Sqrt[-((a-b-c) (a+b-c) (a-b+c) (a+b+c))]):>4Area

Factor[Together//@TrilinearEquation@@v]

eqn=Factor[ ((a^12) (α^2)-32 (a^8) (Area^2) (α^2)+256 (a^4) (Area^4) (α^2)-3 (a^10) (b^2) (α^2)-8 (a^8) Area (b^2) (α^2)+64 (a^6) (Area^2) (b^2) (α^2)+128 (a^4) (Area^3) (b^2) (α^2)-256 (a^2) (Area^4) (b^2) (α^2)+2 (a^8) (b^4) (α^2)+24 (a^6) Area (b^4) (α^2)-32 (a^4) (Area^2) (b^4) (α^2)-128 (a^2) (Area^3) (b^4) (α^2)+2 (a^6) (b^6) (α^2)-24 (a^4) Area (b^6) (α^2)-3 (a^4) (b^8) (α^2)+8 (a^2) Area (b^8) (α^2)+(a^2) (b^10) (α^2)-3 (a^10) (c^2) (α^2)-8 (a^8) Area (c^2) (α^2)+64 (a^6) (Area^2) (c^2) (α^2)+128 (a^4) (Area^3) (c^2) (α^2)-256 (a^2) (Area^4) (c^2) (α^2)-4 (a^8) (b^2) (c^2) (α^2)-16 (a^6) Area (b^2) (c^2) (α^2)+64 (a^4) (Area^2) (b^2) (c^2) (α^2)-256 (a^2) (Area^3) (b^2) (c^2) (α^2)+14 (a^6) (b^4) (c^2) (α^2)+88 (a^4) Area (b^4) (c^2) (α^2)-4 (a^4) (b^6) (c^2) (α^2)-64 (a^2) Area (b^6) (c^2) (α^2)-3 (a^2) (b^8) (c^2) (α^2)+2 (a^8) (c^4) (α^2)+24 (a^6) Area (c^4) (α^2)-32 (a^4) (Area^2) (c^4) (α^2)-128 (a^2) (Area^3) (c^4) (α^2)+14 (a^6) (b^2) (c^4) (α^2)+88 (a^4) Area (b^2) (c^4) (α^2)+14 (a^4) (b^4) (c^4) (α^2)+112 (a^2) Area (b^4) (c^4) (α^2)+2 (a^2) (b^6) (c^4) (α^2)+2 (a^6) (c^6) (α^2)-24 (a^4) Area (c^6) (α^2)-4 (a^4) (b^2) (c^6) (α^2)-64 (a^2) Area (b^2) (c^6) (α^2)+2 (a^2) (b^4) (c^6) (α^2)-3 (a^4) (c^8) (α^2)+8 (a^2) Area (c^8) (α^2)-3 (a^2) (b^2) (c^8) (α^2)+(a^2) (c^10) (α^2)+2 (a^11) b α β+8 (a^9) Area b α β-32 (a^7) (Area^2) b α β-128 (a^5) (Area^3) b α β-6 (a^9) (b^3) α β-32 (a^7) Area (b^3) α β+32 (a^5) (Area^2) (b^3) α β+256 (a^3) (Area^3) (b^3) α β+4 (a^7) (b^5) α β+48 (a^5) Area (b^5) α β+32 (a^3) (Area^2) (b^5) α β-128 a (Area^3) (b^5) α β+4 (a^5) (b^7) α β-32 (a^3) Area (b^7) α β-32 a (Area^2) (b^7) α β-6 (a^3) (b^9) α β+8 a Area (b^9) α β+2 a (b^11) α β-6 (a^9) b (c^2) α β-72 (a^7) Area b (c^2) α β-128 (a^5) (Area^2) b (c^2) α β+384 (a^3) (Area^3) b (c^2) α β+512 a (Area^4) b (c^2) α β-8 (a^7) (b^3) (c^2) α β+72 (a^5) Area (b^3) (c^2) α β+512 (a^3) (Area^2) (b^3) (c^2) α β+384 a (Area^3) (b^3) (c^2) α β+28 (a^5) (b^5) (c^2) α β+72 (a^3) Area (b^5) (c^2) α β-128 a (Area^2) (b^5) (c^2) α β-8 (a^3) (b^7) (c^2) α β-72 a Area (b^7) (c^2) α β-6 a (b^9) (c^2) α β+4 (a^7) b (c^4) α β+136 (a^5) Area b (c^4) α β+352 (a^3) (Area^2) b (c^4) α β+256 a (Area^3) b (c^4) α β+28 (a^5) (b^3) (c^4) α β+176 (a^3) Area (b^3) (c^4) α β+352 a (Area^2) (b^3) (c^4) α β+28 (a^3) (b^5) (c^4) α β+136 a Area (b^5) (c^4) α β+4 a (b^7) (c^4) α β+4 (a^5) b (c^6) α β-88 (a^3) Area b (c^6) α β-192 a (Area^2) b (c^6) α β-8 (a^3) (b^3) (c^6) α β-88 a Area (b^3) (c^6) α β+4 a (b^5) (c^6) α β-6 (a^3) b (c^8) α β+16 a Area b (c^8) α β-6 a (b^3) (c^8) α β+2 a b (c^10) α β+(a^10) (b^2) (β^2)+8 (a^8) Area (b^2) (β^2)-128 (a^4) (Area^3) (b^2) (β^2)-256 (a^2) (Area^4) (b^2) (β^2)-3 (a^8) (b^4) (β^2)-24 (a^6) Area (b^4) (β^2)-32 (a^4) (Area^2) (b^4) (β^2)+128 (a^2) (Area^3) (b^4) (β^2)+256 (Area^4) (b^4) (β^2)+2 (a^6) (b^6) (β^2)+24 (a^4) Area (b^6) (β^2)+64 (a^2) (Area^2) (b^6) (β^2)+2 (a^4) (b^8) (β^2)-8 (a^2) Area (b^8) (β^2)-32 (Area^2) (b^8) (β^2)-3 (a^2) (b^10) (β^2)+(b^12) (β^2)-3 (a^8) (b^2) (c^2) (β^2)-64 (a^6) Area (b^2) (c^2) (β^2)-256 (a^2) (Area^3) (b^2) (c^2) (β^2)-256 (Area^4) (b^2) (c^2) (β^2)-4 (a^6) (b^4) (c^2) (β^2)+88 (a^4) Area (b^4) (c^2) (β^2)+64 (a^2) (Area^2) (b^4) (c^2) (β^2)+128 (Area^3) (b^4) (c^2) (β^2)+14 (a^4) (b^6) (c^2) (β^2)-16 (a^2) Area (b^6) (c^2) (β^2)+64 (Area^2) (b^6) (c^2) (β^2)-4 (a^2) (b^8) (c^2) (β^2)-8 Area (b^8) (c^2) (β^2)-3 (b^10) (c^2) (β^2)+2 (a^6) (b^2) (c^4) (β^2)+112 (a^4) Area (b^2) (c^4) (β^2)-128 (Area^3) (b^2) (c^4) (β^2)+14 (a^4) (b^4) (c^4) (β^2)+88 (a^2) Area (b^4) (c^4) (β^2)-32 (Area^2) (b^4) (c^4) (β^2)+14 (a^2) (b^6) (c^4) (β^2)+24 Area (b^6) (c^4) (β^2)+2 (b^8) (c^4) (β^2)+2 (a^4) (b^2) (c^6) (β^2)-64 (a^2) Area (b^2) (c^6) (β^2)-4 (a^2) (b^4) (c^6) (β^2)-24 Area (b^4) (c^6) (β^2)+2 (b^6) (c^6) (β^2)-3 (a^2) (b^2) (c^8) (β^2)+8 Area (b^2) (c^8) (β^2)-3 (b^4) (c^8) (β^2)+(b^2) (c^10) (β^2)+2 (a^11) c α γ+8 (a^9) Area c α γ-32 (a^7) (Area^2) c α γ-128 (a^5) (Area^3) c α γ-6 (a^9) (b^2) c α γ-72 (a^7) Area (b^2) c α γ-128 (a^5) (Area^2) (b^2) c α γ+384 (a^3) (Area^3) (b^2) c α γ+512 a (Area^4) (b^2) c α γ+4 (a^7) (b^4) c α γ+136 (a^5) Area (b^4) c α γ+352 (a^3) (Area^2) (b^4) c α γ+256 a (Area^3) (b^4) c α γ+4 (a^5) (b^6) c α γ-88 (a^3) Area (b^6) c α γ-192 a (Area^2) (b^6) c α γ-6 (a^3) (b^8) c α γ+16 a Area (b^8) c α γ+2 a (b^10) c α γ-6 (a^9) (c^3) α γ-32 (a^7) Area (c^3) α γ+32 (a^5) (Area^2) (c^3) α γ+256 (a^3) (Area^3) (c^3) α γ-8 (a^7) (b^2) (c^3) α γ+72 (a^5) Area (b^2) (c^3) α γ+512 (a^3) (Area^2) (b^2) (c^3) α γ+384 a (Area^3) (b^2) (c^3) α γ+28 (a^5) (b^4) (c^3) α γ+176 (a^3) Area (b^4) (c^3) α γ+352 a (Area^2) (b^4) (c^3) α γ-8 (a^3) (b^6) (c^3) α γ-88 a Area (b^6) (c^3) α γ-6 a (b^8) (c^3) α γ+4 (a^7) (c^5) α γ+48 (a^5) Area (c^5) α γ+32 (a^3) (Area^2) (c^5) α γ-128 a (Area^3) (c^5) α γ+28 (a^5) (b^2) (c^5) α γ+72 (a^3) Area (b^2) (c^5) α γ-128 a (Area^2) (b^2) (c^5) α γ+28 (a^3) (b^4) (c^5) α γ+136 a Area (b^4) (c^5) α γ+4 a (b^6) (c^5) α γ+4 (a^5) (c^7) α γ-32 (a^3) Area (c^7) α γ-32 a (Area^2) (c^7) α γ-8 (a^3) (b^2) (c^7) α γ-72 a Area (b^2) (c^7) α γ+4 a (b^4) (c^7) α γ-6 (a^3) (c^9) α γ+8 a Area (c^9) α γ-6 a (b^2) (c^9) α γ+2 a (c^11) α γ+2 (a^10) b c β γ+16 (a^8) Area b c β γ-192 (a^6) (Area^2) b c β γ+256 (a^4) (Area^3) b c β γ+512 (a^2) (Area^4) b c β γ-6 (a^8) (b^3) c β γ-88 (a^6) Area (b^3) c β γ+352 (a^4) (Area^2) (b^3) c β γ+384 (a^2) (Area^3) (b^3) c β γ+4 (a^6) (b^5) c β γ+136 (a^4) Area (b^5) c β γ-128 (a^2) (Area^2) (b^5) c β γ-128 (Area^3) (b^5) c β γ+4 (a^4) (b^7) c β γ-72 (a^2) Area (b^7) c β γ-32 (Area^2) (b^7) c β γ-6 (a^2) (b^9) c β γ+8 Area (b^9) c β γ+2 (b^11) c β γ-6 (a^8) b (c^3) β γ-88 (a^6) Area b (c^3) β γ+352 (a^4) (Area^2) b (c^3) β γ+384 (a^2) (Area^3) b (c^3) β γ-8 (a^6) (b^3) (c^3) β γ+176 (a^4) Area (b^3) (c^3) β γ+512 (a^2) (Area^2) (b^3) (c^3) β γ+256 (Area^3) (b^3) (c^3) β γ+28 (a^4) (b^5) (c^3) β γ+72 (a^2) Area (b^5) (c^3) β γ+32 (Area^2) (b^5) (c^3) β γ-8 (a^2) (b^7) (c^3) β γ-32 Area (b^7) (c^3) β γ-6 (b^9) (c^3) β γ+4 (a^6) b (c^5) β γ+136 (a^4) Area b (c^5) β γ-128 (a^2) (Area^2) b (c^5) β γ-128 (Area^3) b (c^5) β γ+28 (a^4) (b^3) (c^5) β γ+72 (a^2) Area (b^3) (c^5) β γ+32 (Area^2) (b^3) (c^5) β γ+28 (a^2) (b^5) (c^5) β γ+48 Area (b^5) (c^5) β γ+4 (b^7) (c^5) β γ+4 (a^4) b (c^7) β γ-72 (a^2) Area b (c^7) β γ-32 (Area^2) b (c^7) β γ-8 (a^2) (b^3) (c^7) β γ-32 Area (b^3) (c^7) β γ+4 (b^5) (c^7) β γ-6 (a^2) b (c^9) β γ+8 Area b (c^9) β γ-6 (b^3) (c^9) β γ+2 b (c^11) β γ+(a^10) (c^2) (γ^2)+8 (a^8) Area (c^2) (γ^2)-128 (a^4) (Area^3) (c^2) (γ^2)-256 (a^2) (Area^4) (c^2) (γ^2)-3 (a^8) (b^2) (c^2) (γ^2)-64 (a^6) Area (b^2) (c^2) (γ^2)-256 (a^2) (Area^3) (b^2) (c^2) (γ^2)-256 (Area^4) (b^2) (c^2) (γ^2)+2 (a^6) (b^4) (c^2) (γ^2)+112 (a^4) Area (b^4) (c^2) (γ^2)-128 (Area^3) (b^4) (c^2) (γ^2)+2 (a^4) (b^6) (c^2) (γ^2)-64 (a^2) Area (b^6) (c^2) (γ^2)-3 (a^2) (b^8) (c^2) (γ^2)+8 Area (b^8) (c^2) (γ^2)+(b^10) (c^2) (γ^2)-3 (a^8) (c^4) (γ^2)-24 (a^6) Area (c^4) (γ^2)-32 (a^4) (Area^2) (c^4) (γ^2)+128 (a^2) (Area^3) (c^4) (γ^2)+256 (Area^4) (c^4) (γ^2)-4 (a^6) (b^2) (c^4) (γ^2)+88 (a^4) Area (b^2) (c^4) (γ^2)+64 (a^2) (Area^2) (b^2) (c^4) (γ^2)+128 (Area^3) (b^2) (c^4) (γ^2)+14 (a^4) (b^4) (c^4) (γ^2)+88 (a^2) Area (b^4) (c^4) (γ^2)-32 (Area^2) (b^4) (c^4) (γ^2)-4 (a^2) (b^6) (c^4) (γ^2)-24 Area (b^6) (c^4) (γ^2)-3 (b^8) (c^4) (γ^2)+2 (a^6) (c^6) (γ^2)+24 (a^4) Area (c^6) (γ^2)+64 (a^2) (Area^2) (c^6) (γ^2)+14 (a^4) (b^2) (c^6) (γ^2)-16 (a^2) Area (b^2) (c^6) (γ^2)+64 (Area^2) (b^2) (c^6) (γ^2)+14 (a^2) (b^4) (c^6) (γ^2)+24 Area (b^4) (c^6) (γ^2)+2 (b^6) (c^6) (γ^2)+2 (a^4) (c^8) (γ^2)-8 (a^2) Area (c^8) (γ^2)-32 (Area^2) (c^8) (γ^2)-4 (a^2) (b^2) (c^8) (γ^2)-8 Area (b^2) (c^8) (γ^2)+2 (b^4) (c^8) (γ^2)-3 (a^2) (c^10) (γ^2)-3 (b^2) (c^10) (γ^2)+(c^12) (γ^2))/.Area^n_.:>Area^Mod[n,2]ToSidelengths[Area]^(n-Mod[n,2])]

eqn=((a^8) (α^2)-2 (a^6) (b^2) (α^2)-4 (a^4) Area (b^2) (α^2)+(a^4) (b^4) (α^2)+4 (a^2) Area (b^4) (α^2)-2 (a^6) (c^2) (α^2)-4 (a^4) Area (c^2) (α^2)-2 (a^4) (b^2) (c^2) (α^2)-8 (a^2) Area (b^2) (c^2) (α^2)+(a^4) (c^4) (α^2)+4 (a^2) Area (c^4) (α^2)+(a^7) b α β+4 (a^5) Area b α β-(a^5) (b^3) α β-8 (a^3) Area (b^3) α β-(a^3) (b^5) α β+4 a Area (b^5) α β+a (b^7) α β+2 (a^5) b (c^2) α β-20 (a^3) Area b (c^2) α β-12 (a^3) (b^3) (c^2) α β-20 a Area (b^3) (c^2) α β+2 a (b^5) (c^2) α β-7 (a^3) b (c^4) α β-7 a (b^3) (c^4) α β+4 a b (c^6) α β+4 (a^4) Area (b^2) (β^2)+(a^4) (b^4) (β^2)-4 (a^2) Area (b^4) (β^2)-2 (a^2) (b^6) (β^2)+(b^8) (β^2)-8 (a^2) Area (b^2) (c^2) (β^2)-2 (a^2) (b^4) (c^2) (β^2)-4 Area (b^4) (c^2) (β^2)-2 (b^6) (c^2) (β^2)+4 Area (b^2) (c^4) (β^2)+(b^4) (c^4) (β^2)+(a^7) c α γ+4 (a^5) Area c α γ+2 (a^5) (b^2) c α γ-20 (a^3) Area (b^2) c α γ-7 (a^3) (b^4) c α γ+4 a (b^6) c α γ-(a^5) (c^3) α γ-8 (a^3) Area (c^3) α γ-12 (a^3) (b^2) (c^3) α γ-20 a Area (b^2) (c^3) α γ-7 a (b^4) (c^3) α γ-(a^3) (c^5) α γ+4 a Area (c^5) α γ+2 a (b^2) (c^5) α γ+a (c^7) α γ+4 (a^6) b c β γ-7 (a^4) (b^3) c β γ-20 (a^2) Area (b^3) c β γ+2 (a^2) (b^5) c β γ+4 Area (b^5) c β γ+(b^7) c β γ-7 (a^4) b (c^3) β γ-20 (a^2) Area b (c^3) β γ-12 (a^2) (b^3) (c^3) β γ-8 Area (b^3) (c^3) β γ-(b^5) (c^3) β γ+2 (a^2) b (c^5) β γ+4 Area b (c^5) β γ-(b^3) (c^5) β γ+b (c^7) β γ+4 (a^4) Area (c^2) (γ^2)-8 (a^2) Area (b^2) (c^2) (γ^2)+4 Area (b^4) (c^2) (γ^2)+(a^4) (c^4) (γ^2)-4 (a^2) Area (c^4) (γ^2)-2 (a^2) (b^2) (c^4) (γ^2)-4 Area (b^2) (c^4) (γ^2)+(b^4) (c^4) (γ^2)-2 (a^2) (c^6) (γ^2)-2 (b^2) (c^6) (γ^2)+(c^8) (γ^2));

circ=Circumcircle[eqn];

Symmetrize[Center[circ][[1]]]

((Factor[-2 (a^4)+3 (a^2) (b^2)-(b^4)+3 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]+4 Area((b^2)+(c^2)))/a)

IncidentCenters[KimberlingCenters,ToSidelengths[CyclicTrilinears[((-2 (a^4)+3 (a^2) (b^2)-(b^4)+3 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)+4 Area ((b^2)+(c^2)))/a)]]]//Timing

CenterName[X[641]]

CenterName[X[485]]

CenterFunction[X[641]]//InputForm

Symmetrize[CircleFunction[eqn][[1]]]

a (Factor[(a^6)-2 (a^4) (b^2)+(a^2) (b^4)-2 (a^4) (c^2)-2 (a^2) (b^2) (c^2)+(a^2) (c^4)]+Area Factor[-4 (a^2)  (b^2)+4  (b^4)-4 (a^2)  (c^2)-8  (b^2) (c^2)+4  (c^4)])

Factor[ToAngles[%]/.(a-b-c) (a+b-c) (a-b+c) (a+b+c):>-(4Area^2)]

InputForm[%]

IncidentCenters[KimberlingCenters,ToSidelengths[CyclicTrilinears[a ((a^6)-2 (a^4) (b^2)+4 (a^2) Area (b^2)+(a^2) (b^4)-4 Area (b^4)-2 (a^4) (c^2)+4 (a^2) Area (c^2)-2 (a^2) (b^2) (c^2)+8 Area (b^2) (c^2)+(a^2) (c^4)-4 Area (c^4))]]]//Timing

FullSimplify[Factor[Together//@ToSidelengths[Distance[Trilinears[X[641]],TrilinearVertexMatrix[OuterVectenTriangle][[1]]]^2]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]/.Sqrt[x_]:>FullSimplify[Sqrt[x],TriangleQ[]]],TriangleQ[]]

((a-b-c) (a+b-c) (a-b+c) (a+b+c) (Factor[2 (a^6)+2 (b^6)-3 (b^4) (c^2)-3 (b^2) (c^4)+2 (c^6)-3 (a^4) ((b^2)+(c^2))-(a^2) (3 (b^4)+14 (b^2) (c^2)+3 (c^4))]-5(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) ((a^2) (b^2) +(a^2) (c^2) + (b^2) (c^2) )))/(2 (((Factor[-2 (a^4)-2 ((((b^2)-(c^2)))^2)+4 (a^2) ((b^2)+(c^2))]+ (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])((a^2)+(b^2)+(c^2))))^2))

InputForm[Sqrt[%]]

CircleRadius[OuterVectenCircle]//TraditionalForm

(1/(Sqrt[2]))(√((-16Area^2(2 (a^6)-3 (b^2) (a^4)-3 (c^2) (a^4)-3 (b^4) (a^2)-3 (c^4) (a^2)-14 (b^2) (c^2) (a^2)+2 (b^6)+2 (c^6)-3 (b^2) (c^4)-3 (b^4) (c^2)-5  4 Area  ((a^2) (b^2)+(c^2) (b^2)+(a^2) (c^2))))/(((4 Area ((a^2)+(b^2)+(c^2))-2 (-16 Area^2)))^2)))//TraditionalForm

((2 (Sqrt[2]) )/(Abs[(32 Δ+4 ((a^2)+(b^2)+(c^2)) )]))√(-( (2 (a^6)-3 (b^2) (a^4)-3 (c^2) (a^4)-3 (b^4) (a^2)-3 (c^4) (a^2)-14 (b^2) (c^2) (a^2)+2 (b^6)+2 (c^6)-3 (b^2) (c^4)-3 (b^4) (c^2)-20 Δ ((a^2) (b^2)+(c^2) (b^2)+(a^2) (c^2)))))

(1/(2Abs[((a^2)+(b^2)+(c^2))+8 Δ]))( (Sqrt[2]) √(-2 (a^6)+3 (a^4) (b^2)+3 (a^2) (b^4)-2 (b^6)+3 (a^4) (c^2)+14 (a^2) (b^2) (c^2)+3 (b^4) (c^2)+3 (a^2) (c^4)+3 (b^2) (c^4)-2 (c^6)+20 ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2)) Δ))//TraditionalForm

InputForm[%56]

CircleRadius[OuterVectenCircle[t]]//N

(1/((Sqrt[2]) Abs[(a^2)+(b^2)+(c^2)+8 Δ]))(√(-2 (a^6)+3 (b^2) (a^4)+3 (c^2) (a^4)+3 (b^4) (a^2)+3 (c^4) (a^2)+14 (b^2) (c^2) (a^2)-2 (b^6)-2 (c^6)+3 (b^2) (c^4)+3 (b^4) (c^2)+20 ((a^2) (b^2)+(c^2) (b^2)+(a^2) (c^2)) Δ))/.Δ:>ToSidelengths[Area]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]//N