ConcurrentMedians::usage="Show that the medians are concurrent in a tetrahedron.The medians start from the barycenter of a triangular face to the opposite vertex.The medians intersect at the center of gravity of the tetrahedron. In this example, an irregular tetrahedron was chosen to show that this property is valid for any tetrahedron, it is known as Commandino theorem.Reference: Personal.";
Development :
Barycenter of each face:
In[]:=
P1={1,1,1};P2={3,6,0};P3={0,3,2};P4={2,3,7};====
Barycenter
123
P1+P2+P3
3
Barycenter
124
P1+P2+P4
3
Barycenter
134
P1+P3+P4
3
Barycenter
234
P2+P3+P4
3
,,1;
4
3
10
3
2,,;
10
3
8
3
1,,;
7
3
10
3
,4,3;
5
3
Director vector of each median:
In[]:=
DirectorVector4=P4-DirectorVector3=P3-DirectorVector2=P2-DirectorVector1=P1-
Barycenter
123
Barycenter
124
Barycenter
134
Barycenter
234
,-,6;
2
3
1
3
-2,-,-;
1
3
2
3
2,,-;
11
3
10
3
-,-3,-2;
2
3
Medians point of intersection (tetrahedron centroid):
In[]:=
Solve====,====,{x,y,z}Solve====,====,{x,y,z}
x-
4
3
2
3
y-
10
3
-
1
3
z-1
6
x-2
-2
y-
10
3
-
1
3
z-
8
3
-
2
3
x-1
2
y-
7
3
11
3
z-
10
3
-
10
3
x-
5
3
-
2
3
y-4
-3
z-3
-2
x,y,z;
3
2
13
4
5
2
x,y,z;
3
2
13
4
5
2
Medians point of intersection (tetrahedron centroid) by formula:
In[]:=
G=
P1+P2+P3+P4
4
,,;
3
2
13
4
5
2
In[]:=
gr=Show[{Graphics3D[{Red,Ball[G,0.06],Black,Ball[,0.04],Ball[,0.04],Ball[,0.04],Ball[,0.04],Ball[P1,0.04],Text[Style[Subscript[P,1],14,Bold],P1,{0,-1.2}],Ball[P2,0.04],Text[Style[Subscript[P,2],14,Bold],P2,{0,-1.2}],Ball[P3,0.04],Text[Style[Subscript[P,3],14,Bold],P3,{0,-1.2}],Ball[P4,0.04],Text[Style[Subscript[P,4],14,Bold],P4,{0,-1.2}],Red,Thick,Line[{,P4}],Line[{,P3}],Line[{,P2}],Line[{,P1}],Green,Opacity[0.4],EdgeForm[Thick],Tetrahedron[{P1,P2,P3,P4}]}]},AxesLabel->{x,y,z},AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,-2.4,2.},ImageSize700]
Barycenter
123
Barycenter
124
Barycenter
134
Barycenter
234
Barycenter
123
Barycenter
124
Barycenter
134
Barycenter
234
Out[]=
In[]:=
ResourceFunction["SpinShow"][gr,30]
Out[]=
