Orthic Inconic

Author

Eric W. Weisstein
February 8, 2005

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

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

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

Equation

Inconic parameters

CenterFunction[Orthocenter]

{x,y,z}={Cos[A],Cos[B],Cos[C]};

Trilinear Equations

eqn1=TrilinearEquation[OrthicInconic]

FullSimplify[eqn1]

CyclicSum[a^2 SA^2 α^2 - 2 b c SB SC β γ]//TraditionalForm

eqn2=Factor[ToAngles[CyclicSum[a^2 SA^2 α^2 - 2 b c SB SC β γ]]/(a b c)^2]

eqn1==eqn2

Center

DistanceTrilinears[{c y+b z,a z+c x,b x+a y}]//Simplify

CenterFunction[SymmedianPoint]

Brianchon Point

1/{x,y,z}

CenterFunction[Orthocenter]

Figure

t=Triangle[{{0,0},{1,0},{.15,.8}}];
bp=Orthocenter[t];
g1=Show[Graphics[{
t,
{Red,OrthicInconic[t]},
{
Blue,
LongestLine[Coordinates/@{bp,#},1]&/@VertexPoints[t],
PerpSquare[#,.03,1]&/@Transpose[{Line/@Transpose[Vertices/@{ReferenceTriangle[t],OrthicTriangle[t]}],Sides[t]}]
},
{
Green,
OrthicTriangle[t]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[OrthicTriangle[t]],{"(Subscript[H, A])","(Subscript[H, B])","(Subscript[H, C])"}}],
{
{X[125][t],"(Subscript[X, 125])",Red},
{SymmedianPoint[t],"K",Red},
{Orthocenter[t],"H",Blue}
}
],
{PointSize[.025],LabelOffset->-.04}
]
}],
PlotRange->{{-.1,1.1},{-.1,.9}},AspectRatio->Automatic]

t=Triangle[{{0,0},{1,0},{-.15,.8}}];
bp=Orthocenter[t];
g2=Show[Graphics[{
{
Purple,
Sides[t,2]
},
t,
{Red,OrthicInconic[t]},
{
Blue,
LongestLine[Coordinates/@{bp,#},1]&/@VertexPoints[t],
PerpSquare[#,.05,1]&/@Transpose[{Line/@Transpose[Vertices/@{ReferenceTriangle[t],OrthicTriangle[t]}],Sides[t]}]
},
{
Green,
OrthicTriangle[t]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[OrthicTriangle[t]],{"(Subscript[H, A])","(Subscript[H, B])","(Subscript[H, C])"}}],
{
{X[125][t],"(Subscript[X, 125])",{Red,LabelOffset->{.08,.05}}},
{SymmedianPoint[t],"K",Red},
{Orthocenter[t],"H",Blue}
}
],
{PointSize[.025],LabelOffset->-.06}
]
}],
PlotRange->{{-.7,1.3},{-.5,1.1}},AspectRatio->Automatic]

Show[GraphicsArray[{g1,g2}]]

Incident Centers

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

First/@ctrs

Length[ctrs]

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

Check

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