Euclidean Plane is Abstract Geometry

Theorem

The Euclidean plane $\struct {\R^2, L_E}$ is an abstract geometry.

We will show that the axioms for an abstract geometry hold.

Let $P = \tuple {x_1, y_1}$ and $Q = \tuple {x_2, y_2}$ be two distinct points in $\struct {\R^2, L_E}$.

If $x_1 = x_2 = a$ then $P, Q \in L_a$.

If $x_1 \ne x_2$ then let:

$m = \dfrac {y_2 - y_1} {x_2 - x_1}$

$b = y_2 - m x_2$

Then $P, Q \in L_{m,b}$, since:

$b + m x_1 = y_2 - m \paren {x_2 - x_1} = y_2 - \paren {y_2 - y_1} = y_1$

$b + m x_2 = y_2 - m \paren {x_2 - x_2} = y_2$

So any two points in $\R^2$ lie on a line in $L_E$.

$\Box$

Let $a \in \R$ be arbitrary.

Then:

$\tuple {a, 0}, \tuple {a, 1} \in L_a$

Let $m, b \in \R$ be arbiitrary.

Then:

$\tuple {0, b}, \tuple {1, m + b} \in L_{m, b}$

So every line in $L_E$ has at least two distinct points.

$\Box$

Hence $\struct {\R^2, L_E}$ is an abstract geometry.

$\blacksquare$