BisectionTheorem::usage="If a plane passes through a bimedian (segment that joins the midpoints of two opposite edges) of a tetraherdon, divide its volume into two equal parts. In this example, an irregular tetrahedron was chosen to show that this property is valid for any tetrahedron. Reference: Personal.";
Development :
Midpoints of two opposite edges:
In[]:=
P1={1,1,1};P2={3,6,0};P3={0,3,2};P4={2,3,7};​​M1=
P1+P4
2
​​M2=
P2+P3
2
​​

3
2
,2,4;

3
2
,
9
2
,1;
Plane that passes through the bimedian and through point P2:
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(M2-M1)(P2-M1)
2,-
9
2
,-
15
4
;
2x-
9
2
y-
15
4
z+D==0;​​2(3)-
9
2
(6)-
15
4
(0)+D==0;​​D=21;​​8x-18y-15z+84==0;
Calculation of volumes:
In[]:=
P
1
P
2
P
3
P
4
tetrahedron==
1
6
Abs[Det[{{1,1,1,1},{3,6,0,1},{0,3,2,1},{2,3,7,1}}]]​​
M
1
P
2
P
3
P
4
tetrahedron==
1
6
AbsDet
3
2
,2,4,1,{3,6,0,1},{0,3,2,1},{2,3,7,1}​​
M
1
P
1
P
2
P
3
tetrahedron==
1
6
AbsDet
3
2
,2,4,1,{1,1,1,1},{3,6,0,1},{0,3,2,1}
tetrahedron
P
1
P
2
P
3
P
4

59
6
;
tetrahedron
M
1
P
2
P
3
P
4

59
12
;
tetrahedron
M
1
P
1
P
2
P
3

59
12
;
These calculations verify the theorem
In[]:=
gr=Show[{ContourPlot3D[8x-18y-15z+84==0,{x,0,5},{y,0,6},{z,-5,5},MeshFalse,ContourStyle{Yellow,Opacity[0.1],Specularity[White,40]}],​​Graphics3D[{Red,Ball[M1,0.08],​​Text[Style[Subscript[M,1],14,Bold],M1,{0,-1.2}],​​Ball[M2,0.08],​​Text[Style[Subscript[M,2],14,Bold],M2,{0,-1.2}],Black,​​Ball[P1,0.08],​​Text[Style[Subscript[P,1],14,Bold],P1,{0,-1.2}],​​Ball[P2,0.08],​​Text[Style[Subscript[P,2],14,Bold],P2,{0,-1.2}],​​Ball[P3,0.08],​​Text[Style[Subscript[P,3],14,Bold],P3,{0,-1.2}],​​Ball[P4,0.08],​​Text[Style[Subscript[P,4],14,Bold],P4,{0,-1.2}],Red,Thick,​​Line[{M1,M2}],​​Green,Opacity[0.4],EdgeForm[Thick],​​Tetrahedron[{P1,P2,P3,P4}]}]},AxesLabel->{x,y,z},AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,-2.4,2.},ImageSize700]
Out[]=
In[]:=
ResourceFunction["SpinShow"][gr,30]
Out[]=