I have a three pairs of points namely : $p_{1,1}, p_{1,2}, p_{2,1}, p_{2,2}, p_{3,1}, p_{3,2}$, I want to find $p$ such that : $|p-p_{i,1}| = |p-p_{i,2}|$ for all $i$.
I know that these represent three planes, but I am not sure whether these planes do meet in a single point. My simple question is even if the three planes are not parallel do they always meet in a single point. Intuitively I can visualize three planes which are not parallel but don't meet at a single point but how do I know that mathematically ?