identity prove?

Question

Let f be an isometry, and let a and b be two distinct points where
f(a)=a and f(b)=b. Prove that f is either the identity or a reflection.

f is in the plane

Answer 1

This is only true in the plane. In space, f could be a rotation about the axis ab.

I will prove it for the plane: without loss of generality we may assume that a = (0,0) and b = (1,0). Let c = (x,y), and f(c) = (x',y'). Then

d(a,c)^2 = x^2 + y^2
d(b,c)^2 = (x-1)^2 + y^2 = (x^2 + y^2) + (1 - 2x)

d(f(a),f(c))^2 = x'^2 + y'^2
d(f(b),f(c))^2 = (x'-1)^2 + y'^2 = (x'^2 + y'^2) + (1 - 2x')

It follows that
x'^2 + y'^2 = x^2 + y^2;
1 - 2x = 1 - 2x'
x = x'
y = +- y'

Showing the either f: (x,y) -> (x,y) is the identity;
or f: (x,y) -> (x,-y) is the reflection in the x-axis (the line ab).

Answer 2

on which space?
it is important to know the domain of f,
otherwise it is false

Answer 3

here you are
http://www-maths.swan.ac.uk/pgrads/an/ito.pdf#search=%22f%20isometry%20%22f(a)%3Da%22%20%22f(b)%3Db%22%22