Quasigroup from a finite projective plane order 2

Question

I have only seen a quasigroup in terms of a latin square with numbers and so am not sure where a quasigroup comes up in the circumstance of a projective plane. Could someone provide an example of a quasigroup from the picture below?

Answer 1

There is an example that appears here https://en.wikipedia.org/wiki/Steiner_system#Steiner_triple_systems:

The points $P_i$ are the elements of the quasigroup and multiplication is defined to be $P_i*P_i=P_i$ and $P_i*P_j$ is the third point on the line for $i\neq j$ (note that this point is unique).

However, this construction of a quasigroup follows from the picture being a Steiner triple, and not from being projective plane. I think you need to provide more details on the question for a better answer.