We can see that the following proposition is true.
Proposition : Each triangle $ABD, ACD, BCD$ is a right triangle for $$A(0,b,0), B(a,0,0), C(0,0,0)\ \ \ (a\gt 0, b\gt 0)$$ $\iff D$ is either $(a,b,0)\ \text{or}\ (a,0,p)\ \text{or}\ (0,b,p)\ \text{or}\ (q,0,r)\ \text{or}\ (0,s,t)$ where $$p\not=0, q^2+r^2=qa, 0\lt q\lt a, s^2+t^2=sb, 0\lt s\lt b.$$
Using this proposition enables us to show the following theorem.
Theorem : There is no set of five distinct points within three-dimensional space such that every set of three points of them are the vertices of a right triangle.
Then, here is my question.
Question : Can you show the theorem without using the proposition?