1. The problem statement, all variables and given/known data Suppose f is an isometry that fixes O (origin). Prove f preserves midpoints of line segments. 3. The attempt at a solution Geometricallly, f could be a reflection in which case it would not preserve the mid point of any line segment that does not intersect the origin anywhere. So I don't see a proof at all and infact sees a mistake.
That is true. I was thinking along the wrong lines (no pun intended) in that I was thinking that f maps midpoint to the exact same mid point. Everything makes geometric sense. The only problem is to prove it algebraically. Can't see how to do it.
I have worked out the quesion in the OP. I now need to show that f(ru)=rf(u) with the same conditions given in the OP.