Parry Reflection Point

Author

Eric W. Weisstein
November 15, 2005

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

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

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

Derivation

lmn=ParallelLine[#,EulerLine]&/@IdentityMatrix[3];

llmmnn=Reflection@@@Transpose[{lmn,TrilinearLine/@IdentityMatrix[3]}];

t=Triangle[{{0,0},{1,0},{.35,.6}}];
Show[Graphics[{
{Yellow,Sides[t,1]},
t,
{Blue,#[t]&/@lmn},
{Green,#[t]&/@llmmnn},
{Red,EulerLine[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{{X[399][t],"(Subscript[X, 399])",Green}}
],PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->{{-1,2},{-1,1.5}}]

Concurrent

Expand[ToSidelengths[ConcurrentQ[llmmnn]]]//Timing

Together//@ToSidelengths[ConcurrentQ[llmmnn]]//Timing

Factor[ToSidelengths[ConcurrentQ[llmmnn]]]//Timing

Symmetrize[ToSidelengths[Intersections@@Take[llmmnn,2]]]

Trilinears[X[399]]

Symmetrize[ToSidelengths[Trilinears[X[399]]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]]]