Given two points $A$ and $B$ and two distances $m$ and $n$, find a point that has distance $m$ fom $A$ and $n$ from $B$

Question

I know that, as long as the distance from $|GI|<m+n$, as you can see in the figure $1$, I can constructo such point by the intersection of the circles with center at $G$ and radius $m$ and with center at $I$ and radius $n$. Therefore, there is another condition, ilustrated in figure $2$ that I cannot understand what is the limitation. Could somebody help me? Maybe there is a relation to $|m-n|$ but I cannot visualize.

Answer 1

The second diagram satisfies the condition you found from Figure 1, yet the two circles don't intersect.

Given any two points $A$ and $B$ and any $m,n \in \mathbb R$, suppose that there exists some point $C$ such that $|AC| = m$ and $|BC| = n$. Then by the Triangle Inequality, we know that the sum of any two sides in the triangle $ABC$ must exceed that of the third side so that:

\begin{align*} m + n &> |AB| \tag 1\\ m + |AB| &> n \tag 2\\ |AB| + n &> m \tag 3 \end{align*}

Figure 2 illustrates what goes wrong when condition $(2)$ is violated.

Answer 2

the condition is $GI \ge |m-n|$. This is because of the reversed triangle inequality:

$$ GI \ge |GA - GB| $$or, in an Euclidian space: $$ |x-y|\ge \left||x| - |y|\right| $$