Evans Conic

Author

Eric W. Weisstein
January 29, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.2,.7}}];
Show[Graphics[{
t,
{
Red,
EvansConic[t]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Red]&/@{
{FermatPoint[1][t],"X"},
{FermatPoint[2][t],"X'"},
{NapoleonPoint[1][t],"N"},
{NapoleonPoint[2][t],"N'"},
{IsodynamicPoint[1][t],"S"},
{IsodynamicPoint[2][t],"S'"}
}
],
{PointSize[.02],LabelOffset->{0,-.05}}
]
}],
PlotRange->{{-.1,1.1},{-.1,.75}},AspectRatio->Automatic]

Equation

Factor[ToSidelengths[TrilinearEquation[EvansConic]]]//TraditionalForm

Factor[Coefficient[ToSidelengths[TrilinearEquation[EvansConic]],#]&/@{(α^2),(β^2),(γ^2),α β,β γ,γ α}]

Factor[ToAngles[%]]

From Peter Moses, Jan. 12, 2005

Simplify[With[{SA = b c Cos[A],
SB = a c Cos[B],
SC =a b Cos[C],
S=2Area
},
(SB - SC)^2 (SA^2 - 3 S^2) (3 SA^2 - S^2) α^2a^2 + 2 (SA - SB) (SC - SA) (3a^4 SA^2 + (a^2 SA + SB SC) (SB^2 + SC^2 - 6 SB SC)) β γ b c]]

(a^2) (3 ((Subscript[S, A])^2)-(S^2)) ( ((Subscript[S, A])^2)-3 (S^2)) ((( (Subscript[S, B])- (Subscript[S, C])))^2) (α^2)+2 b c ( (Subscript[S, C])- (Subscript[S, A])) ( (Subscript[S, A])- (Subscript[S, B])) (3  ((Subscript[S, A])^2) (a^4)+( (Subscript[S, A]) (a^2)+ (Subscript[S, B])  (Subscript[S, C])) ( ((Subscript[S, B])^2)-6  (Subscript[S, C])  (Subscript[S, B])+S ((Subscript[S, C])^2))) β γ

eqn=Factor[-a b c ToSidelengths[With[{
SA = (1/2) (-(a^2)+(b^2)+(c^2)),
SB = (1/2) ((a^2)-(b^2)+(c^2)),
SC = (1/2) ((a^2)+(b^2)-(c^2)),
S=2Area
},
-1/(a b c)Plus@@CyclicTrilinears[(SB - SC)^2 (SA^2 - 3 S^2) (3 SA^2 - S^2) α^2a^2 + 2 (SA - SB) (SC - SA) (3a^4 SA^2 + (a^2 SA + SB SC) (SB^2 + SC^2 - 6 SB SC)) β γ b c]
]]]

(Factor/@Coefficient[eqn,#]&/@{(α^2),(β^2),(γ^2),α β,α γ,β γ})

Center

ctr=Symmetrize/@ToSidelengths[Center[EvansConic]]

Symmetrize[-FullSimplify[ToSidelengths[12Area/a-a Cot[BrocardAngle]],TriangleQ[]]]

IncidentCenters[KimberlingCenters,ctr]//Timing

Incident Points

(ctrs=IncidentCenters[KimberlingCenters,TrilinearEquation[EvansConic]])//Timing

Length[ctrs]

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

Check

Print[{#,Factor[ToSidelengths[TrilinearEquation[EvansConic]/.Thread[{α,β,γ}->Trilinears[#]]/.trig:(_Sin|_Cos|_Csc|_Sec):>TrigExpand[trig]]/.Sqrt[x_]:>FullSimplify[Sqrt[ExpandAll[x]],TriangleQ[]]]//Timing}]&/@ctrs;

Derivation Attempt

eqn=Conic[Trilinears/@{FermatPoint[1],FermatPoint[2],NapoleonPoint[1],NapoleonPoint[2],IsodynamicPoint[1]}];

eqn2=TrigExpand[eqn];//Timing

ByteCount/@{eqn,eqn2}/10^6.

ByteCount[eqn3=eqn/.f:(_Sin|_Cos|_Tan|_Sec|_Csc|_Cot):>TrigExpand[f]]/10^6.//Timing

ByteCount[eqn4=Factor[ExpandAll[eqn3]]]/10^6.//Timing

Simplify[Expand[Numerator[Together[Expand[eqn/.f:(_Sin|_Cos|_Tan|_Csc):>TrigExpand[f]]]]]]//Timing

Simplify[Expand[Numerator[Together[Expand[eqn/.f:(_Sin|_Cos|_Tan|_Csc):>TrigExpand[f]]]]/.C->Pi-A-B/.f:(_Sin|_Cos):>TrigExpand[f]]]//Timing

FullSimplify[%[[2]],TriangleQ[]]//Timing