Draw a line (considered endless in both directions), put a point somewhere on each side of the line. Label these points A and B.
Use a geometric construction to locate a point, P, on the line, which is equidistant from A and B.

two points plus one line



Can this point P always be found for any position of A or B ?
If you believe that this is true how might you construct a proof?
If it is false, identify the circumstances when no point P, equidistant from A and B, exists.

Draw a new arrangement of one line and two points, one on each side of the line.
Imagine creating a collection of similar arrangements.
Can you suggest useful parameters which would uniquely define or identify each arrangement?
Parameters are the numbers ( measurements or ratios ) you might communicate to another person, say over the telephone, if you wanted them to produce exactly the same arrangement as your own.

These parameters define each arrangement.
Mark a point O somewhere on the line and express the length OP in terms of your parameters.