Questions related to the definition of the smallest affine plane

Here I have two questions related to affine planes.

1. The smallest affine plane has four points and six lines where

$$\mathcal{P}=\{A, B, C, D\}$$

and

$$\mathcal{L}=\{\{AB\}, \{AC\}, \{AD\}, \{BC\}, \{BD\}, \{CD\}\}$$

as illustrated in the following picture. However, looking at the figure, I observe that $l(A,D)$ does not contain the point $C$, but there is no line passing through $C$ and parallel to $l(A,D)$. Then, how can this figure describe the smallest four-points affine plane?

1. How can I show that the axioms for an affine plane hold in case of considering the following four-points set

$$\mathcal{P}=\{(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)\}$$

and the following set of lines

$$\mathcal{L}=\{x=0, y=0, z=0, x=y, x=z, y=z\}$$

?

1. The line $l(B,C)$ passes through $C$ and is parallel to $l(A, D)$. Recall that two lines in an affine plane are parallel if they do not intersect, as is the case here.
• In Fig it looks like that the lines $l(B,C)$ and $l(A,D)$ intersect each other – Noor Aslam Sep 13 '18 at 4:49