Hey guys. I am having extreme dificulty on this question. I am wondering if anyone knows how to go about this, seems some what too difficult for me:
"IF we have two points 'A' and 'B' which are on opposite sides of a straight line 'M'. Locate a point 'X' on 'M' where the difference betwween |AX| and |BX| is a maximum."
What do you mean "opposite sides"? A straight line only has length. Do you mean above and below the line? Or, are 'A' and 'B' endpoints of a line segment? Also, it is conventional in geometry to use lower case letters when designating a line with a single letter.
Restate your argument to clear these questions up.
This is to answer the question and thetwo responses.
Assume we are operating in a plane. Modern axiomatic geometry has axioms about sides of a line.
The axiom vary but go something like this.
Definition: Two points are on different sides of if and only if there is a point of between the points.
Given a line there is a point such .
The is at least .
There is a point such that ( is between ).
Now we have a and .
"IF we have two points 'A' and 'B' which are on opposite sides of a straight line 'M'.
Definition there is a point such that .
I've played a little bit with your problem.
I've noticed that the maximum difference seems to occur if the line m is the angle bisector of the angle . But unfortunately I haven't found a proof for this statement .
If the point X is approaching the "ends" of the line the difference of distances has a limit which is represented by the green line segment.
You are asked to find the point X - and I hope nobody asked you to prove that this point is the only one satisfying all conditions
1. draw the perpendicular bisector of AB which cuts m in M.
2. the difference of distances AM - BM = 0 therefore the differences have a minimum if X is placed on M
3. draw the point A' which is the reflection of A over m.
4. draw the line BA' which cuts m in X. And that's exactly(?) the point you are looking for.