ConcurrentPlanes::usage="Show that the six planes drawn by the midpoints of the edges of a tetrahedron, perpendicular to the opposite edge, pass through a point. That point is known as the point of Monge. References: Usunáriz Balanzategui, Ubaldo; Usunáriz Sala, Ignacio. Problemas de Geometría; Personal.";
Development:
Midpoints of each edge:
In[]:=
P1={1,1,1};P2={3,6,0};P3={0,3,2};P4={2,3,7};​​M1=
P1+P2
2
​​M2=
P2+P4
2
​​M3=
P3+P4
2
​​M4=
P1+P3
2
​​M5=
P1+P4
2
​​M6=
P2+P3
2
2,
7
2
,
1
2
;

5
2
,
9
2
,
7
2
;
1,3,
9
2
;

1
2
,2,
3
2
;

3
2
,2,4;

3
2
,
9
2
,1;
Director vectors of each edge:
In[]:=
P1={1,1,1};P2={3,6,0};P3={0,3,2};P4={2,3,7};​​P4-P3​​P3-P1​​P2-P1​​P4-P2​​P3-P2​​P4-P1
{2,0,5};
{-1,2,1};
{2,5,-1};
{-1,-3,7};
{-3,-3,2};
{1,2,6};
Plane that starts from M1 with director vector the edge
P
3
P
4
:
​​2x+5z+D=0;
2(2)+5
1
2
+D=0;​​D=-
13
2
;​​2x+5z-
13
2
==0;​​4x+10z-13==0;
Plane that starts from M2 with director vector the edge
P
1
P
3
:
-x+2y+z+D=0;​​-
5
2
+2
9
2
+
7
2
+D=0;​​D=-10;​​x-2y-z+10==0;
Plane that starts from M3 with director vector the edge
P
1
P
2
:
2x+5y-z+D=0;​​2(1)+5(3)-
9
2
+D=0;​​D=-
25
2
;​​2x+5y-z-
25
2
==0;​​4x+10y-2z-25==0;
Plane that starts from M4 with director vector the edge
P
2
P
4
:
-x-3y+7z+D=0;​​-
1
2
-3(2)+7
3
2
+D=0;​​D=-4;​​x+3y-7z+4==0;
Plane that starts from M5 with director vector the edge
P
2
P
3
:
-3x-3y+2z+D=0;​​-3
3
2
-3(2)+2(4)+D=0;​​D=
5
2
;​​3x+3y-2z-
5
2
==0;​​6x+6y-4z-5==0;
Plane that starts from M6 with director vector the edge
P
1
P
4
:
x+2y+6z+D==0;​​
3
2
+2
9
2
+6(1)+D=0;​​D=-
33
2
;​​x+2y+6z-
33
2
==0;​​2x+4y+12z-33==0;
Monge point:
In[]:=
Solve[{2x+4y+12z-33==0,6x+6y-4z-5==0,x+3y-7z+4==0},{x,y,z}]
x-
159
118
,y
201
59
,z
217
118
;
Checking on the remaining planes:
In[]:=
Solve[{4x+10z-13==0,x-2y-z+10==0,4x+10y-2z-25==0},{x,y,z}]
x-
159
118
,y
201
59
,z
217
118
;
In[]:=
gr=Showeq1=4x+10z-13;eq2=x-2y-z+10;eq3=x-2y-z+10;eq4=x+3y-7z+4;eq5=6x+6y-4z-5;eq6=2x+4y+12z-33;​​ContourPlot3D[eq1==0,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->{Orange,Opacity[0.3],Specularity[White,40]}],​​ContourPlot3D[eq2==0,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->{Orange,Opacity[0.3],Specularity[White,40]}],​​ContourPlot3D[eq3==0,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->{Orange,Opacity[0.3],Specularity[White,40]}],​​ContourPlot3D[eq4==0,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->{Orange,Opacity[0.3],Specularity[White,40]}],​​ContourPlot3D[eq5==0,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->{Orange,Opacity[0.3],Specularity[White,40]}],​​ContourPlot3D[eq6==0,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->{Orange,Opacity[0.3],Specularity[White,40]}],Graphics3DBlack,Ball-
159
118
,
201
59
,
217
118
,0.1,Ball[{1,1,1},0.08],​​Text[Style[Subscript[P,1],14,Bold],{1,1,1},{0,-1.2}],​​Ball[{3,6,0},0.08],​​Text[Style[Subscript[P,2],14,Bold],{3,6,0},{0,-1.2}],​​Ball[{0,3,2},0.08],​​Text[Style[Subscript[P,3],14,Bold],{0,3,2},{0,-1.2}],​​Ball[{2,3,7},0.08],​​Text[Style[Subscript[P,4],14,Bold],{2,3,7},{0,-1.2}],Green,Opacity[0.4],EdgeForm[Thick],Tetrahedron[{{1,1,1},{3,6,0},{0,3,2},{2,3,7}}],AxesLabel->{x,y,z},AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,-2.4,2.},ImageSize700
Note: This is the form in which the ResourceFunction “Monge” would be made: