VivianiTheorem3D::usage="Viviani's theorem says that in a equilateral triangle the distances of an inner point on each side add up to a length equal to that of its height. Here it will be shown with an example how Viviani's theorem wasalso fulfilled in 3D using a regular tetraherdon. References: Personal, based in Viviani's theorem.";
Development:
In[]:=
ShowGraphics3D[{Green,Opacity[0.05],EdgeForm[Thick],Tetrahedron[{{1,1,1},{1,-1,-1},{-1,1,-1},{-1,-1,1}}]}],Graphics3D[{Black,Thick,Ball[{1,1,1},0.025]}],Graphics3D[{Red,Thick,Ball[{1,-1,-1},0.025]}],Graphics3D[{Blue,Thick,Ball[{-1,1,-1},0.025]}],Graphics3D[{Green,Thick,Ball[{-1,-1,1},0.025]}],Graphics3DPurple,Thick,Ball0,
1
2
,
1
40
,0.025,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[]=
Cartesian equation of the GreenBlueRed plane:
In[]:=
{-1,-1,1};{-1,1,-1};{1,-1,-1};​​{-1,-1,1}+u{0,2,-2}+v{2,0,-2};​​Eliminate[{x==-1+2v,y==-1+2u,z==1-2u-2v},{u,v}]
-1-y-zx;​​x+y+z+1==0;
Cartesian equation of the GreenRedBlack plane:
In[]:=
{-1,-1,1};{1,-1,-1};{1,1,1};​​{-1,-1,1}+u{2,0,-2}+v{2,2,0};​​Eliminate[{x==-1+2u+2v,y==-1+2v,z==1-2u},{u,v}]
1+y-zx;​​x-y+z-1==0;
Cartesian equation of the GreenBlackBlue plane:
In[]:=
{-1,-1,1};{1,1,1};{-1,1,-1};​​{-1,-1,1}+u{2,2,0}+v{0,2,-2};​​Eliminate[{x==-1+2u,y==-1+2u+2v,z==1-2v},{u,v}]
-1+y+zx;​​x-y-z+1==0;
Cartesian equation of the RedBlackBlue plane:
In[]:=
{1,-1,-1};{1,1,1};{-1,1,-1};​​{1,-1,-1}+u{0,2,2}+v{-2,2,0};​​Eliminate[{x==1-2v,y==-1+2u+2v,z==-1+2u},{u,v}]​​
1-y+zx;​​x+y-z-1==0;
Tetrahedron height (distance from the green point to the RedBlackBlue plane):
In[]:=
h=
Abs[-1-1-1-1]
1+1+1
4
3
;
Distance from the purple inner point to the GreenBlueRed plane:
In[]:=
d1=
Abs0+
1
2
+
1
40
+1
3
61
40
3
;
Distance from the purple inner point to the GreenRedBlack plane:
In[]:=
d2=
Abs0-
1
2
+
1
40
-1
3
59
40
3
;
Distance from the purple inner point to the GreenBlackBlue plane:
In[]:=
d3=
Abs0-
1
2
-
1
40
+1
3
19
40
3
;
Distance from the purple inner point to the RedBlackBlue plane:
In[]:=
d4=
Abs0+
1
2
-
1
40
-1
3
7
3
40
;
Sum of the four distances:
In[]:=
61
40
3
+
59
40
3
+
19
40
3
+
7
3
40
//FullSimplify
4
3
;
The result of the sum coincides with the height of the tetrahedron, Viviani’s 3D theorem is fulfilled.
Calculations needed to draw the final graph:
In[]:=
Solvex+y+z+1==0,
x-0
1
==
y-
1
2
1
==
z-
1
40
1
,{x,y,z}
x-
61
120
,y-
1
120
,z-
29
60
;
In[]:=
Solvex-y+z-1==0,
x-0
1
==
y-
1
2
-1
==
z-
1
40
1
,{x,y,z}
x
59
120
,y
1
120
,z
31
60
;
In[]:=
Solvex-y-z+1==0,
x-0
1
==
y-
1
2
-1
==
z-
1
40
-1
,{x,y,z}
x-
19
120
,y
79
120
,z
11
60
;
In[]:=
Solvex+y-z-1==0,
x-0
1
==
y-
1
2
1
==
z-
1
40
-1
,{x,y,z}