Points are encoded in homogeneous coordinates; think of the tacking on another coordinate, i.e., (x,y,1). So, when going from Euclidean to PGA: (x,y) \rightarrow x e_{20} + y e_{01} + 1 e_{12}.
When going from PGA to Euclidean: x' e_{20} + y' e_{01} + w' e_{21} \rightarrow (x,y) = (\frac{x'}{w'}, \frac{y'}{w'}), i.e., the ‘weight’ w factor of e_{12} needs to be normalized to 1. The reverse mapping is sometimes called de-homogenizing. Consequently, all scalar multiples of any PGA point \alpha p, where \alpha \in \mathbb{R}, \; p = x' e_{20} + y' e_{01} + w' e_{12} represent the same Euclidean point.