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F preserves midpoint?

by pivoxa15
Tags: midpoint, preserves
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Jan26-08, 05:46 AM
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.
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Jan26-08, 05:58 AM
P: 13
But in the case of a reflection the transformation of a midpoint is still a midpoint, no?
Jan26-08, 06:06 AM
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.

Jan26-08, 12:17 PM
P: 139
F preserves midpoint?

Are we in R^n?
Jan26-08, 03:47 PM
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.
Jan26-08, 07:13 PM
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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