UsoDeMonge::usage="Obtain the Monge point for the next tetrahedron:, , 0, 2, , 0, , 3, 0 y , , 2 , then trace the crcumscribed sphere.The Monge point of a tetrahedron is the point of concurrence of the anti-mediator planes that pass through the midpoint of each edge and is perpendicular to the opposite edge.Referencia: Wolfram MathWorld: Monge Point; Personal."
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Development:
In[]:=
ρ=ResourceFunction["Monge"],,0,2,,0,,3,0,,,2
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Out[]=
,,,Hyperplane,-2,0,3,,1,Hyperplane1,-,0,,3,1,Hyperplane{2,-1,-2},,,0,Hyperplane-,,0,,2,1,Hyperplane-,1,-2,4,,0,Hyperplane1,,-2,,,0
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tet=,,0,2,,0,,3,0,,,2;Graphics3D[{Tube[#]&/@Subsets[tet,{2}],Green,Sphere[ρ[[1]],0.05],Opacity[.4],Red,Polygon/@Subsets[tet,{3}],Yellow,Opacity[.2],ρ[[2]]}]
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Out[]=
Barycenter coordinates:
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G=
,,0+2,,0+,3,0+,,2
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,,
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Obtaining the coordinates of the circumcenter::
Knowing that the barycenter is the midpoint of the segment whose ends are the circumcenter and the point of Monge, we have::
Circumcenter=2G-M;Circumcenter=2,,-,,
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,,
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Radius of the sphere:
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r=Norm,,-,,0
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In[]:=
r=Norm,,-2,,0
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In[]:=
r=Norm,,-,3,0
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In[]:=
r=Norm,,-,,2
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What we wanted to show..
Full graph:
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Showρ=ResourceFunction["Monge"],,0,2,,0,,3,0,,,2;tet=,,0,2,,0,,3,0,,,2;Graphics3D[{Tube[#]&/@Subsets[tet,{2}],Green,Sphere[ρ[[1]],0.08],Opacity[.4],Red,Polygon/@Subsets[tet,{3}],Yellow,Opacity[.2],ρ[[2]]}],Graphics3DBlue,Opacity[0.2],Sphere,,,,Graphics3DBlack,Ball,,,0.08,Graphics3DBlack,Ball,,,0.08
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Out[]=
