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Figure
1% fig-iii-20.tex — III.20: inscribed angle theorem.
2\begin{figure}[H]
3\centering
4\begin{tikzpicture}[scale=1.1, line cap=round]
5 \coordinate (O) at (0, 0);
6 \def\r{2}
7 \draw[thin] (O) circle (\r);
8 \coordinate (A) at ({\r*cos(150)}, {\r*sin(150)}); % on circle, left
9 \coordinate (B) at ({\r*cos(30)}, {\r*sin(30)}); % on circle, right
10 \coordinate (P) at ({\r*cos(270)}, {\r*sin(270)}); % on circle, bottom (point opposite arc)
11 % Chord AB.
12 \draw[very thick] (A) -- (B);
13 % Inscribed angle from P.
14 \draw[thick] (P) -- (A);
15 \draw[thick] (P) -- (B);
16 % Central angle from O.
17 \draw[thick, dashed] (O) -- (A);
18 \draw[thick, dashed] (O) -- (B);
19 \node[below] at (O) {$O$};
20 \node[above left] at (A) {$A$};
21 \node[above right] at (B) {$B$};
22 \node[below] at (P) {$P$};
23 % Indicate angles.
24 \draw[->, thin] (1.1, 0.6) arc[start angle=30, end angle=150, radius=1.2];
25 \node at (0, 1.3) {$2\theta$};
26 \draw[->, thin] (P) ++(60:0.7) arc[start angle=60, end angle=120, radius=0.7];
27 \node at (0, -1.0) {$\theta$};
28\end{tikzpicture}
29\caption{Proposition III.20. The central angle $\angle AOB$ is twice
30the inscribed angle $\angle APB$ subtending the same arc $AB$.
31Corollary: all inscribed angles on the same arc are equal.}
32\label{fig:III.20}
33\end{figure}
34