Algebraic proof for sphere/circle overlap formula

Question

Two spheres or circles denoted by center position vector and radius $ p_0, r_0$ and $p1, r_1$ will overlap if $$ |p_0-p_1| < r_0+r_1$$ I understand geometrically why it works, but how would one derive this formula using only algebra, using for example the equation of the sphere : $|c-x|^2=r^2$ ?

Answer 1

Say that the two spheres have equations: $$ \left\{ \begin{align} \big| x - p_0 \big| &= r_0 \\ \big| x - p_1 \big| &= r_1 \end{align} \right. $$

Then, the triangle inequality yields $$ \big| p_0 - p_1 \big| \le \big| x - p_0 \big| + \big| x - p_1 \big| = r_0 + r_1 $$ if and only if such a point $x$ exists on both spheres simultaneously.