Bevan Point

Author

Eric W. Weisstein
January 15, 2005

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

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

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

Plot

t=Triangle[{{0,0},{1,0},{.3,.6}}];
Show[Graphics[{
t,
{Blue,Excircles[t]},
{Green,ex=ExcentralTriangle[t]},
{Red,Circumcircle[ex]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[ex],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
{
{BevanPoint[t],"V",Red}
}
],PointSize[.025]]
}],
AspectRatio->Automatic,PlotRange->All]

Reflection of Incenter in Circumcenter

t=Triangle[{{0,0},{1,0},{.3,.6}}];
Show[Graphics[{
t,
{Red,
l=Line[Coordinates/@{Incenter[t],BevanPoint[t]}],
EqualSideTicksAboutMidpoint[l,1,.015]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{BevanPoint[t],"V",Red},
{Incenter[t],"I",DarkGreen},
{Circumcenter[t],"O",Blue}
}
],PointSize[.025],LabelOffset->.03]
}],
AspectRatio->Automatic,PlotRange->All]

io=FullSimplify[DistanceTrilinears[Distance[Incenter,Circumcenter]],TriangleQ[]]

ov=FullSimplify[DistanceTrilinears[Distance[BevanPoint,Circumcenter]],TriangleQ[]]

io==ov

d=-((a^3)-(a^2) (b+c)+(((b-c))^2) (b+c)-a ((b^2)-3 b c+(c^2)))

Simplify[d-((-a+b+c)(a-b+c)(a+b-c)-a b c)]

8Semiperimeter[]Inradius[]^2

Simplify[d-(8Semiperimeter[]Inradius[]^2-a b c)]

Circumradius[]

FullSimplify[Circumradius[] Sqrt[1-8Semiperimeter[] Inradius[]^2/(a b c)]-io,TriangleQ[]]

R (Sqrt[1-((8 s (r^2))/(a b c))])

FullSimplify[ Circumradius[]^2 8Semiperimeter[] Inradius[]^2/(a b c),TriangleQ[]]

FullSimplify[Sqrt[Circumradius[]^2-a b c/(a+b+c)]-io,TriangleQ[]]

Midpoint of Na-L

t=Triangle[{{0,0},{1,0},{.3,.6}}];
Show[Graphics[{
t,
{Red,
l=Line[Coordinates/@{NagelPoint[t],DeLongchampsPoint[t]}],
EqualSideTicksAboutMidpoint[l,1,.015]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{BevanPoint[t],"V",Red},
{NagelPoint[t],"Na",DarkGreen},
{DeLongchampsPoint[t],"L",Blue}
}
],PointSize[.025],LabelOffset->.03]
}],
AspectRatio->Automatic,PlotRange->All]

nv=FullSimplify[Together//@DistanceTrilinears[Distance[NagelPoint,BevanPoint]],TriangleQ[]]

lv=FullSimplify[Together//@DistanceTrilinears[Distance[DeLongchampsPoint,BevanPoint]],TriangleQ[]]

nv==lv

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

t1=Expand[(a+b+c)^3]

t2=Expand[(-a+b+c)(a-b+c)(a+b-c)]

Reflection of H in Sp

t=Triangle[{{0,0},{1,0},{.3,.6}}];
Show[Graphics[{
t,
{Red,
l=Line[Coordinates/@{Orthocenter[t],BevanPoint[t]}],
EqualSideTicksAboutMidpoint[l,1,.015]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{BevanPoint[t],"V",Red},
{SpiekerCenter[t],"Sp",DarkGreen},
{Orthocenter[t],"H",Blue}
}
],PointSize[.025],LabelOffset->.03]
}],
AspectRatio->Automatic,PlotRange->All]

hsp=FullSimplify[Together//@DistanceTrilinears[Distance[Orthocenter,SpiekerCenter]],TriangleQ[]]

spv=FullSimplify[Together//@DistanceTrilinears[Distance[BevanPoint,SpiekerCenter]],TriangleQ[]]

hsp==spv

t=Expand[(a^4)+(a^2) b c-(a^3) (b+c)-a (((b-c))^2) (b+c)+(((b-c))^2) ((b^2)+b c+(c^2))]

Expand[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]

Expand[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]

Expand[(a+b)(b+c)(a+c)]

V and I Equidistance from the Euler Line

t=Triangle[{{0,0},{1,0},{-.1,.7}}];
lines=PerpendicularLine[#[t],EulerLine[t]]&/@{X[1],X[40]};
Show[Graphics[{
t,
{Blue,EulerLine[t]},
{Red,lines,
PerpSquare[{EulerLine[t],#},.03,1]&/@lines
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{X[1][t],"I",Blue},
{X[40][t],"V",Blue}
}
],
LabelOffset->.05
]
}],AspectRatio->Automatic,PlotRange->{{-.2,1.1},{-.1,.9}}]

Refine[Map[Factor,Together//@ToSidelengths[Distance[EulerLine,X[1]]],3],TriangleQ[]]

Factor//@FullSimplify[%,TriangleQ[]]

Refine[Map[Factor,Together//@ToSidelengths[Distance[EulerLine,X[40]]],3],TriangleQ[]]

Factor//@FullSimplify[%,TriangleQ[]]

Together//@ToSidelengths[(a - b) (b - c) (c - a) (a + b + c) / (4 Distance[Circumcenter,Orthocenter] S)]

Kimberling

CenterName[X[40]]

CenterFunction[X[40]]

Structures Passing through It

pt=BevanPoint;

Circles

First/@Select[{#,IncidentCenters[{pt},#]}&/@Circles,Last[#]=!={}&]

Circle Centers

First/@Select[{#,IncidentCenters[{pt},CircleCenter[#]]}&/@Circles,Last[#]=!={}&]

Conics

First/@Select[{#,IncidentCenters[{pt},#]}&/@Conics,Last[#]=!={}&]

Cubics

First/@Select[{#,IncidentCenters[{pt},#]}&/@TriangleCubics,Last[#]=!={}&]

Lines

First/@Select[{#,IncidentCenters[{pt},#]}&/@Lines,Last[#]=!={}&]