Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

12.6: Axiom III

  • Page ID
    23659
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    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

    \(\measuredangle_h\:(P,Q,R)\mapsto\measuredangle_h P Q R\) 

    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.