Prasolov Point

Author

Eric W. Weisstein
November 13, 2005

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

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

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

Figure

?EqualSideTicksAboutMidpoint

t=Triangle[{{0,0},{1,0},{1/2-1/100,3/4}}];
Show[Graphics[{
t,
{Green,ot=OrthicTriangle[t]},
{Red,rt=Cartesian[Reflection[OrthicTriangle,NinePointCenter][t]]},
{Blue,LongestLine[Append[#,Coordinates[X[68][t]]]]&/@Transpose[Vertices/@{t,rt}]},
{Orange,l=Line/@Transpose[Vertices/@{ot,rt}],
MapIndexed[EqualSideTicksAboutMidpoint[#1,#2[[1]],.015]&,l]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[rt],{"A'","B'","C'"}}],
Append[#,Green]&/@Transpose[{VertexPoints[ot],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"}}],
{{NinePointCenter[t],"Ni",Orange}},
{{X[68][t],"(Subscript[X, 68])",Blue}}
],LabelOffset->.03,PointSize[.022]]
}],AspectRatio->Automatic,PlotRange->All]

Derivation

tri=Reflection[OrthicTriangle,NinePointCenter]

Together//@(ToSidelengths[PerspectiveQ[tri,Triangle[ReferenceTriangle]]/.c_Cos:>TrigExpand[c]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[Expand[s t]],TriangleQ[]])

Symmetrize[ToAngles[Symmetrize[Together//@(ToSidelengths[Perspector[tri,Triangle[ReferenceTriangle]]/.c_Cos:>TrigExpand[c]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[Expand[s t]],TriangleQ[]])]]]

Trilinears[X[68]]