Let be given tetrahedron ABCD, where AB = BC = AC = a, AD = d, AD = e, CD = f. I know that, If the measure of angle of AB and CD equal to Pi/3, then we have d^2 - e^2 - a*f = 0. I tried:
ListTools[Categorize];
L := []; 
for a to 30 do for d to 30 do
for e to 30 do for f to 30 do
if abs(d-e) < a and a < d+e and abs(a-e) < d and d < a+e and abs(d-a) < e and e < d+a and abs(d-f) < a and a < d+f and abs(a-f) < d and d < a+f and abs(d-a) < f and f < d+a and abs(e-f) < a and a < e+f and abs(a-f) < e and e < a+f and abs(a-e) < a and a < a+e and -a*f+d^2-e^2 = 0 and igcd(a, d, e, f) = 1 and nops({a, d, e, f}) = 4
then L := [op(L), [a, d, e, f]] end if end do end do end do end do; 
nops(L); 
L;

Another way to find the length of edges of a tetrahedron knowing that the mesure angle of two opposite?

I want to find the maximize and minimize of the function
f:=x->(cos(x)+sqrt(3)*sin(x))/(cos(x)+sin(x)+2);
I tried 
minimize(f(x), x, location = 'true');
and
maximize(f(x), x, location = 'true');
But I didn't get the results.  How do I find the maximize and minimize of above funciton?

How can I check six numbers: a, b, c, d, e, f are length six sides of a tetrahedron?

I have a list L:=[[0,0,0], [1,0,0], [1,1,0], [0,1,0], [0,0,1], [1,0,1], [1,1,1], [0,1,1]] (8 vertices of a cube). How can I select four vertices of the list to make a regular tetrahedron?

I see from here http://www.mapleprimes.com/questions/220829-How-Do-I-Write-The-Equation-Of-The-Plane to write the equation of a sphere. I tried 
restart; L := [[[0, 0, 0], [2, 0, 0], [0, 2, 0], [2, 2, 0]], [[0, 0, 0], [2, 0, 0], [0, 2, 0], [0, 0, 2]], [[0, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 2]], [[0, 0, 0], [2, 0, 0], [0, 2, 0], [0, 2, 2]], [[0, 0, 0], [2, 0, 0], [0, 2, 0], [2, 2, 2]], [[0, 0, 0], [2, 0, 0], [2, 2, 0], [0, 0, 2]], [[0, 0, 0], [2, 0, 0], [2, 2, 0], [2, 0, 2]], [[0, 0, 0], [2, 0, 0], [2, 2, 0], [0, 2, 2]], [[0, 0, 0], [2, 0, 0], [2, 2, 0], [2, 2, 2]], [[0, 0, 0], [2, 0, 0], [0, 0, 2], [2, 0, 2]], [[0, 0, 0], [2, 0, 0], [0, 0, 2], [0, 2, 2]], [[0, 0, 0], [2, 0, 0], [0, 0, 2], [2, 2, 2]], [[0, 0, 0], [2, 0, 0], [2, 0, 2], [0, 2, 2]], [[0, 0, 0], [2, 0, 0], [2, 0, 2], [2, 2, 2]], [[0, 0, 0], [2, 0, 0], [0, 2, 2], [2, 2, 2]], [[0, 0, 0], [0, 2, 0], [2, 2, 0], [0, 0, 2]], [[0, 0, 0], [0, 2, 0], [2, 2, 0], [2, 0, 2]], [[0, 0, 0], [0, 2, 0], [2, 2, 0], [0, 2, 2]], [[0, 0, 0], [0, 2, 0], [2, 2, 0], [2, 2, 2]], [[0, 0, 0], [0, 2, 0], [0, 0, 2], [2, 0, 2]], [[0, 0, 0], [0, 2, 0], [0, 0, 2], [0, 2, 2]], [[0, 0, 0], [0, 2, 0], [0, 0, 2], [2, 2, 2]], [[0, 0, 0], [0, 2, 0], [2, 0, 2], [0, 2, 2]], [[0, 0, 0], [0, 2, 0], [2, 0, 2], [2, 2, 2]], [[0, 0, 0], [0, 2, 0], [0, 2, 2], [2, 2, 2]], [[0, 0, 0], [2, 2, 0], [0, 0, 2], [2, 0, 2]], [[0, 0, 0], [2, 2, 0], [0, 0, 2], [0, 2, 2]], [[0, 0, 0], [2, 2, 0], [0, 0, 2], [2, 2, 2]], [[0, 0, 0], [2, 2, 0], [2, 0, 2], [0, 2, 2]], [[0, 0, 0], [2, 2, 0], [2, 0, 2], [2, 2, 2]], [[0, 0, 0], [2, 2, 0], [0, 2, 2], [2, 2, 2]], [[0, 0, 0], [0, 0, 2], [2, 0, 2], [0, 2, 2]], [[0, 0, 0], [0, 0, 2], [2, 0, 2], [2, 2, 2]], [[0, 0, 0], [0, 0, 2], [0, 2, 2], [2, 2, 2]], [[0, 0, 0], [2, 0, 2], [0, 2, 2], [2, 2, 2]], [[2, 0, 0], [0, 2, 0], [2, 2, 0], [0, 0, 2]], [[2, 0, 0], [0, 2, 0], [2, 2, 0], [2, 0, 2]], [[2, 0, 0], [0, 2, 0], [2, 2, 0], [0, 2, 2]], [[2, 0, 0], [0, 2, 0], [2, 2, 0], [2, 2, 2]], [[2, 0, 0], [0, 2, 0], [0, 0, 2], [2, 0, 2]], [[2, 0, 0], [0, 2, 0], [0, 0, 2], [0, 2, 2]], [[2, 0, 0], [0, 2, 0], [0, 0, 2], [2, 2, 2]], [[2, 0, 0], [0, 2, 0], [2, 0, 2], [0, 2, 2]], [[2, 0, 0], [0, 2, 0], [2, 0, 2], [2, 2, 2]], [[2, 0, 0], [0, 2, 0], [0, 2, 2], [2, 2, 2]], [[2, 0, 0], [2, 2, 0], [0, 0, 2], [2, 0, 2]], [[2, 0, 0], [2, 2, 0], [0, 0, 2], [0, 2, 2]], [[2, 0, 0], [2, 2, 0], [0, 0, 2], [2, 2, 2]], [[2, 0, 0], [2, 2, 0], [2, 0, 2], [0, 2, 2]], [[2, 0, 0], [2, 2, 0], [2, 0, 2], [2, 2, 2]], [[2, 0, 0], [2, 2, 0], [0, 2, 2], [2, 2, 2]], [[2, 0, 0], [0, 0, 2], [2, 0, 2], [0, 2, 2]], [[2, 0, 0], [0, 0, 2], [2, 0, 2], [2, 2, 2]], [[2, 0, 0], [0, 0, 2], [0, 2, 2], [2, 2, 2]], [[2, 0, 0], [2, 0, 2], [0, 2, 2], [2, 2, 2]], [[0, 2, 0], [2, 2, 0], [0, 0, 2], [2, 0, 2]], [[0, 2, 0], [2, 2, 0], [0, 0, 2], [0, 2, 2]], [[0, 2, 0], [2, 2, 0], [0, 0, 2], [2, 2, 2]], [[0, 2, 0], [2, 2, 0], [2, 0, 2], [0, 2, 2]], [[0, 2, 0], [2, 2, 0], [2, 0, 2], [2, 2, 2]], [[0, 2, 0], [2, 2, 0], [0, 2, 2], [2, 2, 2]], [[0, 2, 0], [0, 0, 2], [2, 0, 2], [0, 2, 2]], [[0, 2, 0], [0, 0, 2], [2, 0, 2], [2, 2, 2]], [[0, 2, 0], [0, 0, 2], [0, 2, 2], [2, 2, 2]], [[0, 2, 0], [2, 0, 2], [0, 2, 2], [2, 2, 2]], [[2, 2, 0], [0, 0, 2], [2, 0, 2], [0, 2, 2]], [[2, 2, 0], [0, 0, 2], [2, 0, 2], [2, 2, 2]], [[2, 2, 0], [0, 0, 2], [0, 2, 2], [2, 2, 2]], [[2, 2, 0], [2, 0, 2], [0, 2, 2], [2, 2, 2]], [[0, 0, 2], [2, 0, 2], [0, 2, 2], [2, 2, 2]]];
getEq := proc (L1::listlist) local p, S, expr; seq(geom3d:-point(p || j, L1[j]), j = 1 .. 4); geom3d:-Equation(geom3d:-sphere(S, [p1, p2, p3, p4], [x, y, z])) end proc; map(getEq, L);

But I couldn't get the result. How can get the result?