Do Euclidean geometry preservers parallelism of lines and area ratios?

Question

Do the Euclidean geometry preserves the properties parallelism of lines and area ratios for any possible transformation?

I know that the Affine geometry do and I think that Euclidean geometry also do it.

What do you mean by "Euclidean geometry"? You seem to be talking about some kind of plane/space transformations, but that isn't clear. — Wojowu, Jan 17 '16 at 12:31
Euclidean space/geometry uses several forms of coordinates. The most prominent is the Cartesian coordinate systems. The group of Euclidean transformations consists of translation, rotation, reflection and scaling. The question is if the the properties of parallelism of lines and area ratios for any of those transformations is preserved? — kotakata, Jan 17 '16 at 12:41
Yes. In fact, areas are preserved, so preservance of ratios is a corollary. It might be of interest to you that all Euclidean transformations are affine. — Wojowu, Jan 17 '16 at 12:43
Oh, I see, Affine geometry is a generalisation of the Euclidean geometry, thank you! — kotakata, Jan 17 '16 at 12:46

score 0 · Answer 1 · answered Jan 17 '16 at 13:32

The usual hierarchy is

Affine transformations (they preserve parallelism and ratios of segments along a line):
- Shear
- Stretch
- Similarity transformations (these also preserve angles and ratios of areas):
  - Dilation (scaling)
  - Euclidean transformations or Isometries (these also preserve distance):
    - Rotation
    - Reflection
    - Translation
    - Glide reflection

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