How to formally define in Euclidean geometry whether two triples of noncollinear points are in the same orientation?

Question

Assume we are working in the Euclidean plane. We are given six points $P, Q, R, P', Q', R'$ such that $P, Q, R$ are noncollinear and $P', Q', R'$ are also noncollinear. How does one formally define the relation "The triples $(P,Q,R)$ and $(P',Q',R')$ have the same orientation". In other words, they are both clockwise or both counterclockwise. Of course, in a general Euclidean plane, clockwise and counterclockwise don't have an absolute meaning. One can only say that two triples of noncollinear points have the same orientation. It is that definition that I am searching for.

Answer 1

Here is one natural way to define it. There is a unique translation $t$ that sends $P$ to $P'$, and then a unique rotation $r$ around $P'$ that sends $t(Q)$ to a point on the ray from $P'$ to $Q'$. Then $(P,Q,R)$ and $(P',Q',R')$ have the same orientation iff $r(t(R))$ is on the same side of line $P'Q'$ as $R$ is.

(The intuition here is that translations and rotations preserve orientation, so you can assume $P=P'$ and $Q$ and $Q'$ are on the same ray from $P$, and then having the orientation is just determined by which side of line $PQ$ the third point is on.)