Basic Examples

Find the Monge point and six midplanes for a tetrahedron:

○
Monge
[{{-1,0,3},{1,-1,1},{2,0,-2},{-2,-2,1}}]
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61
38
,-
63
19
,
43
19
,Hyperplane{-2,1,2},0,-1,-
1
2
,Hyperplane{-3,0,5},-
1
2
,-
3
2
,1,Hyperplane{1,2,2},
3
2
,-
1
2
,-
1
2
,Hyperplane{-1,-1,3},-
3
2
,-1,2,Hyperplane{3,1,0},
1
2
,0,
1
2
,Hyperplane{4,2,-3},0,-
1
2
,2
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A graphic of a tetrahedron, the Monge point, and the six midplanes:

tet={{0,1,-2},{1,3,3},{3,-1,0},{-1,0,0}};​​monge=
○
Monge
[tet];​​Graphics3D[{Tube[#]&/@Subsets[tet,{2}],​​Green,Sphere[monge[[1]],.2],Opacity[.6],​​Red,Polygon/@Subsets[tet,{3}],Yellow,Opacity[.2],monge[[2]]}]
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Scope

The Monge point can be outside of the tetrahedron:

tet={{2,-2,3},{1,0,0},{2,2,1},{3,-1,-2}};​​monge=
○
Monge
[tet];​​Graphics3D[{Tube[#]&/@Subsets[tet,{2}],​​Green,Sphere[monge[[1]],.2],Opacity[.6],​​Red,Polygon/@Subsets[tet,{3}],Yellow,Opacity[.2],monge[[2]]}]
In[]:=
Out[]=