# f preserves midpoint?

by pivoxa15
Tags: midpoint, preserves
 P: 2,268 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.
 P: 13 But in the case of a reflection the transformation of a midpoint is still a midpoint, no?
 P: 2,268 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.
P: 139

## f preserves midpoint?

Are we in R^n?
 P: 13 If f is a isometry, u.v=f(u).f(v) holds. So the vector norm (u.u)^1/2 and distance stays the same.
 P: 2,268 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.

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