Poincaré Plane is Abstract Geometry

Theorem
The Poincaré plane $\left({\mathbb{H}, L_H}\right)$ is an abstract geometry.

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

Axiom 1
Let $P = \left({x_1, y_1}\right)$ and $Q = \left({x_2, y_2}\right)$ be two points in $\left({\mathbb{H}, L_H}\right)$.

If $x_1 = x_2 = a$ then $P,Q \in {}_a L$.

If $x_1 \ne x_2$ then let:


 * $c = \dfrac {y_2^2 - y_1^2 + x_2^2 - x_1^2} {2 \left({x_2 - x_1}\right)}$


 * $r = \sqrt{\left({x_1^2 - c}\right) - y_1^2}$

By definition then $P, Q \in {}_c L_r$.

So any two points in $\mathbb{H}$ lie on a line in $L_H$.

Axiom 2
For any $a \in \R$:


 * $\left({a, 1}\right), \left({a, 2}\right) \in {}_a L$

Also for any $c \in \R$ and $r \in \R_{>0}$:


 * $\displaystyle \left({c + \frac{1}{2} r, \frac{\sqrt{3} }{2} r}\right),\left({c - \frac{1}{2} r, \frac{\sqrt{3} }{2} r}\right) \in {}_c L_r$

So every line in $L_H$ has at least two points.

Hence $\left({\mathbb{H}, L_H}\right)$ is an abstract geometry.