My textbook defines an isometry in the following way:
$$ \text{Let}\quad f:\Pi \rightarrow \Pi \quad \text{be an isometry. Then, there are two possible cases:}$$ $$ \forall A,B,C \in \Pi \quad \angle ABC=\angle f(A)f(B)f(C)$$ In which case the isometry is said to be orientation preserving.
Then, the orientation reversing isometry is defined in a similar way.
I was wondering the following: Suppose that I want to show that an isometry is orientation preserving. it is very clear to me that it is sufficient to simply show that there exists the points A,B,C in the plane that imply $\angle ABC=\angle f(A)f(B)f(C)$. (Note, I understand it is in general wrong to say this, but I suspect that there is a property of isometries that allows you to only have to find one such example of orientation being preserved.) Is my intuition true, and if so, how could one prove my intuition? This is clearly related to the symmetry of the isometry: no point in space is 'special' meaning that there is some sort of symmetry under translation.
It can be proven that any isometry is the composition of 1, 2 or 3 reflections (https://en.wikipedia.org/wiki/Euclidean_plane_isometry#Isometries_as_reflection_group). But under a reflection all triangles change their orientation, so it is not possible for an isometry to change the orientation only of a subset of triangles.