# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

Give the isogonal conjugate of a point with respect to a triangle or tetrahedron

Contributed by:
Ed Pegg Jr

ResourceFunction["IsogonalConjugate"][ gives the isogonal conjugate of point |

The simplex may be a triangle defined by three 2D points or a tetrahedron defined by four 3D points.

Given a triangle Δ*ABC*, the isogonal conjugate of *P* is constructed by reflecting the lines *PA*, *PB* and *PC* about the angle bisectors and taking the intersection.

Given a tetrahedron *ABCD*, the isogonal conjugate of *P* is constructed by reflecting the lines *PA*, *PB*, *PC* and *PD* about the angle bisectors of the dihedral edges and taking the intersection.

In triangle Δ*ABC*, the trilinear coordinates of a point are the ratio *a*:*b*:*c* of signed distances to the sides. The trilinear coordinates of the isogonal conjugate are the ratio *a*^{-1}:*b*^{-1}:*c*^{-1}.

In tetrahedron *ABCD*, the barycentric coordinates of a point are the ratio of signed volumes of the four tetrahedra points by replacing a vertex with the point *a*:*b*:*c**:**d*. Let Δ*A* be the squared face area of Δ*BCD*. The barycentric coordinates of the isogonal conjugate are then Δ*A*/*a*:Δ*B*/*b*:Δ*C*/*c*:Δ*D*/*d*.

Find the isogonal conjugate of a point with respect to a triangle:

In[1]:= |

Out[1]= |

Find the isogonal conjugate of a point with respect to a triangle:

In[2]:= |

Out[4]= |

The isogonal conjugate of the isogonal conjugate is the original point:

In[5]:= |

Out[5]= |

Show the triangle, point (green) and isogonal conjugate (blue):

In[6]:= |

Out[6]= |

The centers of the incircle and excircles are the only four self-conjugate points:

In[7]:= |

Out[7]= |

In[8]:= |

Out[8]= |

In[9]:= |

Out[9]= |

The product of trilinear coordinates for a point and isogonal conjugate gives three identical values:

In[10]:= |

Out[15]= |

Find the Monge point of a semi-random tetrahedron:

In[16]:= |

Out[18]= |

Find the isogonal conjugate of the Monge point:

In[19]:= |

Out[19]= |

The circumsphere of the face reflections of a point is centered at the isogonal conjugate (and vice-versa):

In[20]:= |

Out[22]= |

For a tetrahedron, calculate the five insphere and exsphere points:

In[23]:= |

Out[24]= |

These five points are all equal to their isogonal conjugates:

In[25]:= |

Out[25]= |

Three other points are their own isogonal conjugate in a tetrahedron:

In[26]:= |

Out[27]= |

Show the 16 collinear sets of three points formed by four tetrahedral points and eight self-conjugates:

In[28]:= |

Out[31]= |

Find intersection points and repeat to obtain 50 lines:

In[32]:= |

Out[34]= |

The following triangle has nice Euler line points for the circumcenter, centroid, nine-point center and orthocenter:

In[35]:= |

Out[36]= |

The Jerabek hyperbola is the locus of the isogonal conjugates of points on the Euler line. Use GroebnerBasis to derive the implicit equation:

In[37]:= |

Out[37]= |

The isogonal conjugates of the Euler points are the orthocenter, symmedian, Kosnita point and circumcenter:

In[38]:= |

Out[38]= |

A graphic of the triangle with the Euler line (blue), circumcenter|circumcircle|perpendicular bisectors (red), centroid|medians (cyan), nine-point center|circle (brown) and orthocenter|altitudes (green) and the isogonal conjugate of the Euler line, the Jerabek hyperbola:

In[39]:= |

Out[40]= |

- 1.1.0 – 22 August 2022
- 1.0.0 – 19 August 2022

This work is licensed under a Creative Commons Attribution 4.0 International License