Let A=(0,1) B=(0,0) C=(1,0)
Suppose that f(A) = (0.4,1.8), f(B) = (1,1), and f(C) = (1.8,1.6).
How do we prove that if its not a translate or glide, then its a rotation?
Is it because since glide is a combination of translation and reflection, and this if its not a translation and not a glide, then its also not a reflection. So it has to be a rotation?
You can tell (e.g. by computing the signed area of the triangle) that the transformation $f$ preserves orientation. This rules out reflections and glide reflections. And you can tell it's no translation since the line $AB$ intersects the line $f(A)f(B)$ in a finite point, so $AB\not\Vert f(A)f(B)$ therefore it's no translation either.