Questions related to the definition of the smallest affine plane

Question

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?

2. 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

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

?

Answer 1

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.

For (2) you'll have to be more specific about what you want to know.