There is a line $L$ and two points $A$ and $B$ not lying on it. The points are on the same side of line. A point $C$ is located on the line $L$ such that $|AC - BC|$ is maximised. Determine $C$.
This question was originally asked in a coordinate geometry problem. However, I've ignored numerical values to focus on approach only instead of solution to a particular textbook question.
My approach:
Consider the triangle $∆ABC$. We know that :
$$AC - BC < AB$$
I thought above constraint as the desired maximum value and answered that the maximum value is $AB$. Also, it matched with answer key.
Problem:
The above inequality is strict and does not turn into an equality. If we were to equate them, then the triangle wouldn't exist and $ABC$ would become a line, thus rendering the concept of restraint meaningless.
So, why is my answer correct ( assuming answer key as correct) and how to explain this?
Are there any other and better methods to solve this problem ?
Thanks !