Prove that any transformation of the Euclidean plane which preserves parallelograms can be constructed from the composition of a rigid transformation (an isometry) and two stretches, using synthetic geometry.
This fact is a direct result of the polar decomposition of a matrix. However, this question is asking for a proof using synthetic geometry without coordinates or linear algebra.
(Source: https://mitpress.mit.edu/9780262010146/mathematics/ . They in fact sketch a proof there, using synthetic geometry only, which I'm still trying to deconstruct and fill in.)
We can in fact assert more:
- Stretch 1 must have an invariant line perpendicular to that of Stretch 2
- Any line segment in the plane will be stretched by a scale factor in between that of Stretch 1 and 2
but I have not been able to use either of these facts to help with the proof.