Verifying the Three-Dimensional Analog of the Pythagorean Theorem

Define the distance formula.

> dist:=(p,q)-> sqrt((p[1]-q[1])^2+(p[2]-q[2])^2+(p[3]-q[3])^2);

Define Heron's formula.

> area:=proc(p,q,r) local a,b,c,s; a:=dist(p,q); b:=dist(p,r); c:=dist(q,r); s:=(a+b+c)/2; sqrt(s*(s-a)*(s-b)*(s-c)); end proc;

Give the coordinates of the four vertices of a right tetrahedron.

> p0:=[0,0,0]; p1:=[a,0,0]; p2:=[0,b,0]; p3:=[0,0,c];

Calculate the sum of the squares of the areas of the lateral faces.

> simplify((area(p0,p1,p2))^2+(area(p0,p1,p3))^2+(area(p0,p2,p3))^2);

Calculate the square of the area of the slant face.

> expand((area(p1,p2,p3)^2));

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