Consider the following analytic description of the Beltrami-Cayley-Klein plane: A point is an ordered pair $(x, y)$ such that $x$ and $y$ are real numbers and satisfy $x^2 + y^2 <1.$ A line is any nonempty set of points $(x, y)$ that satisfies some equation $ax + by + c = 0$ such that $a$ and $b$ are real numbers not both zero. Verify betweenness axioms for this model.
For reference purpose, here I provide the betweenness axioms.
B-1: Let $A$ and $B$ be two distinct points. There exists points $C, D,$ and $E$ on $\overleftrightarrow{AB}$ such that $C$ is between $A$ and $B$, $B$ is between $A$ and $D,$ and $A$ is between $E$ and $B.$
B-2: If $A, B,$ and $C$ are points such that $B$ is between $A$ and $C$, then the three points are distinct and on the same line.
B-3: If $A, B,$ and $C$ are points such that $B$ is between $A$ and $C,$ then $B$ is between $C$ and $A.$
B-4: If $A, B,$ and $C$ are three distinct points on the same line, then exactly one of the following statements is true: $B$ is between $A$ and $C,$ $C$ is between $A$ and $B,$ or $A$ is between $C$ and $B.$
B-5: Every line $\ell$ partitions the plane into $\ell$ itself and two convex sets $S_1$ and $S_2$ such that if a point $P$ is in $S_1$ and $Q$ is in $S_2,$ then $\overline{PQ}$ and $\ell$ have a point in common.