Yff Parabola

Author

Eric W. Weisstein
December 13, 2004

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

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

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

Equation

Inconic parameters

{x,y,z}={a(b-c),b(c-a),c(a-b)};

Parabola identity

x/a+y/b+z/c

Trilinear Equations

eqn=x^2 α^2+y^2β^2+z^2γ^2-2y z β γ-2z x γ α-2x y α β

Focus

{a^2/x,b^2/y,c^2/z}

Simplify[Trilinears[X[101]]/((b-c) (a-b) (a-c))]

Brianchon Point

1/{x,y,z}

Select[Centers,Simplify[Subtract@@(First/@ExactTrilinears/@DistanceTrilinears/@CyclicTrilinears/@{CenterFunction[#],1/a/(b-c)}),a>0&&b>0&&c>0]==0&]

CenterFunction[X[86]]

CenterFunction[X[190]]

Figure

t=Triangle[{{0,0},{1,0},{4/10,7/10}}]//N[#,10]&;
Show[Graphics[{
{Purple,Sides[t,5]},
t,
{Red,
g=ParametricPlot[Evaluate[
TrilinearToCartesian[t,{α,#,-1}]&/@(β/.Solve[TrilinearEquation[YffParabola]==0/.γ->-1/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],β])],
{α,0,50},
PlotPoints->100,
DisplayFunction->Identity
][[1]],
Line[#[[1,-1]]&/@Flatten[g]]
},
{
Green,
YffTriangle[t]
},
{Red,LongestLine[Coordinates/@{Orthocenter[t],Mittenpunkt[t]},5]},
{Blue,LongestLine[Append[#,Coordinates[X[190][t]]]]&/@Transpose[Vertices/@{ReferenceTriangle[t],YffTriangle[t]}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[YffTriangle[t]],{"(Subscript[X, A])","(Subscript[X, B])","(Subscript[X, C])"}}],
{
{X[101][t],"(Subscript[X, 101])",Red},
{X[190][t],"(Subscript[X, 190])",{Blue,LabelOffset->{.18,.01}}},
{Mittenpunkt[t],"M",Red},
{Orthocenter[t],"H",Red}
}
],
{PointSize[.025],LabelOffset->.1}
]
}],
PlotRange->{{-.5,2.5},{-.5,2}},
AspectRatio->Automatic]

Incident Centers

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

First/@ctrs

Length[ctrs]

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

{X[#],CenterName[X[#]]}&/@{101}

Check

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