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 $P = \tuple {x_1, y_1}$ and $Q = \tuple {x_2, y_2}$ be two points in $\struct {\H, L_H}$.

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

If $x_1 \ne x_2$ then let:


 * $c = \dfrac {y_2^2 - y_1^2 + x_2^2 - x_1^2} {2 \paren {x_2 - x_1} }$


 * $r = \sqrt{\paren{x_1 - c}^2 + y_1^2}$

From definition of ${}_c L_r$:


 * ${}_c L_r := \set {\tuple {x, y} \in \H : \paren {x - c}^2 + y^2 = r^2}$

We have:

Thus $P, Q \in {}_c L_r$.

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

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


 * $\tuple {a, 1}, \tuple {a, 2} \in {}_a L$

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


 * $\displaystyle \paren {c + \frac{1}{2} r, \frac{\sqrt{3} }{2} r}, \paren {c - \frac{1}{2} r, \frac{\sqrt{3} }{2} r} \in {}_c L_r$

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

Hence $\struct {\H, L_H}$ is an abstract geometry.