12.6: Axiom III

Note that the first part of Axiom III follows directly from the definition of the h-angle measure defined on page . It remains to show that $\measuredangle_h$ satisfies the conditions Axiom IIIa, Axiom IIIb, and Axiom IIIc.

The following two claims say that $\measuredangle_h$ satisfies IIIa and IIIb.

Claim $\PageIndex{1}$

Given an h-half-line $[O P)_h$ and $\alpha\in(-\pi,\pi]$, there is a unique h-half-line $[O Q)_h$ such that $\measuredangle_h P O Q= \alpha$.

Claim $\PageIndex{2}$

For any h-points $P$, $Q$, and $R$ distinct from an h-point $O$, we have

$\measuredangle_h P O Q+\measuredangle_h Q O R \equiv\measuredangle_h P O R.$

Proof of $\PageIndex{1}$ and $\PageIndex{2}$

Applying the main observation, we may assume that $O$ is the center of the absolute. In this case, for any h-point $P \ne O$, the h-half-line $[OP)_h$ is the intersection of the Euclidean half-line $[OP)$ with h-plane. Hence the claim $\PageIndex{1}$ and Claim $\PageIndex{2}$ follow from the axioms Axiom IIIa and Axiom IIIb of the Euclidean plane.

Claim $\PageIndex{3}$

The function

is continuous at any triple of points $(P,Q,R)$ such that $Q\ne P$, $Q\ne R$, and $\measuredangle_h P Q R\ne\pi$.

Proof

Suppose that $O$ denotes the center of the absolute. We can assume that $Q$ is distinct from $O$.

Suppose that $Z$ denotes the inverse of $Q$ in the absolute; suppose that $\Gamma$ denotes the circle perpendicular to the absolute and centered at $Z$. According to Lemma 12.3.1, the point $O$ is the inverse of $Q$ in $\Gamma$.

Let $P'$ and $R'$ denote the inversions in $\Gamma$ of the points $P$ and $R$ respectively. Note that the point $P'$ is completely determined by the points $Q$ and $P$. Moreover, the map $(Q,P)\mapsto P'$ is continuous at any pair of points $(Q,P)$ such that $Q\ne O$. The same is true for the map $(Q,R)\mapsto R'$

According to the main observation

$\measuredangle_h P Q R\equiv -\measuredangle_h P' O R'.$

Since $\measuredangle_h P' O R'=\measuredangle P' O R'$ and the maps $(Q,P)\mapsto P'$, $(Q,R)\mapsto R'$ are continuous, the claim follows from the corresponding axiom of the Euclidean plane.