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 IsotomicConjugate

Source Notebook

Get the isotomic conjugate of a point with respect to a triangle or tetrahedron

Contributed by: Ed Pegg Jr

ResourceFunction["IsotomicConjugate"][simplex,p]

gives the isotomic conjugate of point p with respect to the triangle or tetrahedron simplex.

Details

simplex may be a triangle defined by three 2D points or a tetrahedron defined by four 3D points.
simplex can also be a Triangle, Tetrahedron or Simplex object, while p can be a Point object.
Given a triangle ΔABC, the isotomic conjugate of point P is constructed by reflecting the cevians of PA, PB and PC about the edge midpoints and finding the anticevian.
In triangle ΔABC, the trilinear coordinates of a point are the ratio α:β:γ of signed distances to the sides with side lengths (a,b,c). The trilinear coordinates of the isotomic conjugate are the ratio (a2α)-1:(b2β)-1:(c2γ)-1.
In tetrahedron ABCD, the trilinear coordinates of a point are the ratio α:β:γ:ϵ of signed distances to the faces with areas (a,b,c,d). The trilinear coordinates of the isotomic conjugate are the ratio (a2α)-1:(b2β)-1:(c2γ)-1:(d2ϵ)-1.

Examples

Basic Examples (3) 

Find the isotomic conjugate of an arbitrary point and triangle:

In[1]:=
ResourceFunction["IsotomicConjugate"][ Triangle[{{0, 0}, {1, 0}, {0, 1}}], Point[{2, 2}]] // Simplify
Out[1]=

Find the Gergonne and Nagel points of a triangle:

In[2]:=
tri = {{0, 0}, {4, 0}, {0, 3}}; ge = ResourceFunction["GergonnePoint"][tri][[1]]; na = ResourceFunction["NagelPoint"][tri][[1]]; {ge, na}
Out[5]=

The Gergonne and Nagel points of a triangle are isotomic conjugates of each other:

In[6]:=
{ResourceFunction["IsotomicConjugate"][tri, na], ResourceFunction["IsotomicConjugate"][tri, ge]}
Out[6]=

Show the Gergonne and Nagel points:

In[7]:=
Legended[ Graphics[{Gray, Triangle[tri], Green, InfiniteLine[{ge, #}] & /@ tri, Blue, InfiniteLine[{na, #}] & /@ tri, Disk[ge, .1], Green, Disk[na, .1], Red, Disk[Mean[#], .1] & /@ Subsets[tri, {2}] }], PointLegend[{Green, Blue}, {"Nagel point", "Gergonne point"}]]
Out[7]=

Find the centroid of a tetrahedron:

In[8]:=
tet = {{0, 0, 0}, {0, 0, 6}, {0, 8, 6}, {6, 2, 9}}; centroid = Mean[tet]
Out[9]=

The isotomic conjugate of the centroid is the same point:

In[10]:=
ic = ResourceFunction["IsotomicConjugate"][tet, centroid]
Out[10]=

Scope (2) 

A tetrahedron and its incenter:

In[11]:=
tet = Tetrahedron[{{1, 0, 0}, {1, 0, 1}, {1, 1, 1}, {0, 0, 1}}]; inc = Point[Insphere[tet][[1]] // FullSimplify]
Out[12]=

Compute the isotomic conjugate of the incenter:

In[13]:=
ResourceFunction["IsotomicConjugate"][tet, inc] // Simplify
Out[13]=

Requirements

Wolfram Language 13.0 (December 2021) or above

Version History

  • 1.0.0 – 27 October 2023

Related Resources

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