Euclidean Plane is Abstract Geometry

Theorem

The Euclidean plane $\left({\R^2, L_E}\right)$ is an abstract geometry.

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

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

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:

$b + m x_1 = y_2 - m \left({x_2 - x_1}\right) = y_2 - \left({y_2 - y_1}\right) = y_1$

$b + m x_2 = y_2 - m \left({x_2 - x_2}\right) = y_2$

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

$\Box$

For any $a \in \R$:

$\left({a, 0}\right), \left({a, 1}\right) \in L_a$

Also for any $m, b \in \R$:

$\left({0, b}\right),\left({1, m + b}\right) \in L_{m,b}$

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

$\Box$

Hence $\left({\R^2, L_E}\right)$ is an abstract geometry.

$\blacksquare$