Tixier Point

Author

Eric W. Weisstein
November 15, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
{Yellow,Sides[t,10]},
t,
{Blue,EulerLine[t]},
{Red,KiepertParabola[t]},
{Purple,line=Line[Coordinates/@{X[110][t],X[476][t]}],
EqualSideTicksAboutMidpoint[line,1,.03]},
{Blue,PerpSquare[{line,EulerLine[t]},.05,2]},
{Green,Circumcircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{X[110][t],"(Subscript[X, 110])",Red},
{X[476][t],"(Subscript[X, 476])",Red},
{Circumcenter[t],"O",Green}
}
],PointSize[.03],LabelOffset->.06]
}],AspectRatio->Automatic,PlotRange->{{-.3,1.2},{-.2,.9}},ImageSize->300]

Distance

Simplify[Subtract@@Abs/@Factor[ToSidelengths[Distance[#,EulerLine]/.trig:(_Csc|_Cos|_Sec):>TrigExpand[trig]]/.Sqrt[x_]:>FullSimplify[Sqrt[Expand[x]],TriangleQ[]]&/@{X[110],X[476]}]]

Simplify[Factor[ToSidelengths[Distance[X[476],EulerLine]/.trig:(_Csc|_Cos|_Sec):>TrigExpand[trig]]/.Sqrt[x_]:>FullSimplify[Sqrt[Expand[x]],TriangleQ[]]],TriangleQ[]]

Factor[Simplify[Sqrt[Factor[ToAngles[Factor[%71]^2]]],TriangleQ[]]]

Factor[ToAngles[%82/.Abs->Identity]]

Derivation

Name[X[110]]

CenterFunction[X[476]]

(Sqrt[1-(((((a^2)-(b^2)+(c^2)))^2)/(4 (a^2) (c^2)))])/.Sqrt[x_]:>FullSimplify[Sqrt[Expand[x]],TriangleQ[]]

Factor[(ToSidelengths[TrigExpand[Reflection[Trilinears[X[110]],TrilinearLine[EulerLine]][[1]]]]/.Sqrt[x_]:>FullSimplify[Sqrt[Expand[x]],TriangleQ[]])]

Symmetrize[%]

Symmetrize[ToAngles[%]]

X[476][t]

Cartesian[CyclicTrilinears[(a/(((b^2)-(c^2)) (1+2 Cos[2 A])))][t]]

Factor[ToSidelengths[TrigExpand[Csc[B-C]]]/.Sqrt[x_]:>FullSimplify[Sqrt[Expand[x]],TriangleQ[]]]

Incident On

IncidentCenters[{X[476]},Circumcircle]