Challenge Level

Well done Graham from Feilding High in New
Zealand and also to Andrei from Tudor Vianu National College,
Bucharest, who both gave good answers to the first part of the
problem.

The geometrical solution is to construct the perpendicular bisector of the line $AB$ and mark where it will cross the original line.

This solution works for all cases except where the line $AB$ is itself perpendicular to the original line AND the original line is not the perpendicular bisector of the line $AB$

Can someone now suggest parameters for the general arrangement so that any particular arrangement can be uniquely identified?

The geometrical solution is to construct the perpendicular bisector of the line $AB$ and mark where it will cross the original line.

This solution works for all cases except where the line $AB$ is itself perpendicular to the original line AND the original line is not the perpendicular bisector of the line $AB$

Can someone now suggest parameters for the general arrangement so that any particular arrangement can be uniquely identified?