Incidence axioms for the upper half plane (complex plane).

The axiom I am checking for this question is I1: "For any two distinct points A,B there exist a unique line L containing both points."

Show that I1 is satisfied in the upper half of the complex plane where "lines" are semicircles and x=constant (non-euclidean lines).

I2 and I3 are easily verified using the fact that non-euclidean lines in the complex plane satisfy the equation of the form $(x-a)^2+y^2=r^2$.

Thanks

• what are $I_2$ and $I_3$? – Willemien Dec 15 '15 at 22:22
• The other 2 incidence axioms. – Kevin Dec 15 '15 at 22:23

If $A,B$ have the same $x$-coordinate $x=x_0$, then they both lie on the line $x=x_0$.
If $A,B$ do not have the same $x$-coordinate, let $L$ be the perpendicular bisector of $\overline{AB}$, so $L$ is a non-horizontal line. Let $P=(a,0)$ be the unique point where $L$ intersects the $x$-axis. The distances $d(A,P)$ and $d(B,P)$ are equal (this is true for every point on $L$); set that equal to $r$. Then $A,B$ both lie on the semi-circle $(x-a)^2+y^2=r^2$.