# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

Construct a specified tetrahedron center

Contributed by:
Ed Pegg Jr

ResourceFunction["TetrahedronCenter"][ returns the tetrahedron center identified by " |

For a tetrahedron *ABCD*, various centers corresponding to "*special*" include:

"Incenter" | center of insphere and intersection point of the dihedral angle bisectors |

"Centroid" | center of mass and concurrence of facial centroids |

"Circumcenter" | center of circumsphere and intersection of edge ⟂ bisecting planes |

"Monge" | intersection of midplanes (⟂ to one edge and concurrent with the opposing edge midpoint) |

"12Point" | center of circumsphere for face medians, one-third points to Monge and feet of mongians |

"Symmedian" | point with minimal total squared distance to faces and isogonal conjugate of the centroid |

"Spieker" | incenter of the medial tetrahedron |

"Fermat" | point with minimal total distance to the vertices |

"Parallelians" | point where subplanes parallel to faces have equal area |

"Extangents" | perspector of ABCD and the extangents tetrahedron |

"EulerMedial" | perspector of ABCD and the Euler-medial tetrahedron |

"MedialEuler" | perspector of ABCD and the medial-Euler tetrahedron |

"EulerNegative" | perspector of ABCD and the Euler-negative tetrahedron |

"NegativeEuler" | perspector of ABCD and the negative-Euler tetrahedron |

There are 50,000+ defined triangle centers, but most do not correspond to a tetrahedron center. For example, the altitudes of a tetrahedron do not necessarily intersect.

Find the incenter where the dihedral angle bisectors intersect:

In[1]:= |

Out[2]= |

The incenter is the center of Insphere, tangent to the faces. Show it:

In[3]:= |

Out[3]= |

Find the centroid, which is the center of mass or the concurrence of the face centroids:

In[4]:= |

Out[6]= |

The centroid can also be found with Mean. Show it:

In[7]:= |

Out[7]= |

Find the circumcenter, where the edge-bisecting perpendicular planes intersect:

In[8]:= |

Out[9]= |

The circumcenter is the center of the Circumsphere. Show it:

In[10]:= |

Out[10]= |

In a tetrahedron, a midplane is perpendicular to one edge and concurrent with the midpoint of the opposing edge. Find the Monge point, the concurrence of the midplanes:

In[11]:= |

Out[12]= |

Calculate the cevians of the Monge point:

In[13]:= |

Out[13]= |

Show the Monge point and the cevians:

In[14]:= |

Out[14]= |

Here are the available SubTetrahedron items:

In[15]:= |

For example, here's the reflection tetrahedron, resulting from reflecting each vertex by the opposite face:

In[16]:= |

Out[17]= |

Connecting the vertices shows a perspector point where the tubes intersect, namely the Monge point:

In[18]:= |

Out[18]= |

Various triangle center points, such as the orthocenter, don't always have a tetrahedron center.

For example, in 2D, the Gergonne point is the perspector of the contact triangle, but the lines do not coincide in 3D given the contact tetrahedron:

In[19]:= |

Out[20]= |

The symmedian point has the minimal total distance squared to the faces:

In[21]:= |

Out[22]= |

Check that:

In[23]:= |

Out[24]= |

Show the symmedian point:

In[25]:= |

Out[25]= |

Find the Euler, Euler projected and medial tetrahedra:

In[26]:= |

Out[27]= |

The spheres are all identical. This unique sphere is also known as the 12-point sphere:

In[28]:= |

Out[28]= |

Show the 12-point sphere:

In[29]:= |

Out[29]= |

The Fermat point minimizes the total distance to the vertices:

In[30]:= |

Out[31]= |

Check that:

In[32]:= |

Out[32]= |

Let S be the sum of distances to the Fermat point. The Fermat tetrahedron is a set of segment endpoints S away from the vertices and passing through the Fermat point:

In[33]:= |

Out[33]= |

Circumspheres of the Fermat point and three vertices coincide with the segment endpoints:

In[34]:= |

Out[34]= |

Calculate the Euler-medial tetrahedron and find the Euler-medial point:

In[35]:= |

Out[36]= |

Show that the point is the perspector of the original and Euler-medial tetrahedron:

In[37]:= |

Out[37]= |

Find the parallelians point:

In[38]:= |

Out[39]= |

Move the tetrahedron and calculate the area of the parallel triangles through the point:

In[40]:= |

Out[41]= |

Show the parallelians point and the equal area parallel triangles:

In[42]:= |

Out[42]= |

- 1.0.0 – 31 October 2023

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