//
··
1% fig-iii-31.tex — III.31: angle in a semicircle is right.
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, 0);
9 \coordinate (B) at ( \r, 0);
10 % Diameter AB.
11 \draw[very thick] (A) -- (B);
12 \coordinate (C) at ({\r*cos(70)}, {\r*sin(70)});
13 \draw[very thick] (A) -- (C) -- (B);
14 % Right angle marker at C.
15 \coordinate (Cm1) at ($(C)!0.25!(A)$);
16 \coordinate (Cm2) at ($(C)!0.25!(B)$);
17 \coordinate (Cmid) at ($(Cm1)!0.5!(Cm2)$);
18 \draw[thin] (Cm1) -- ($(Cm1)+(Cm2)-(C)$) -- (Cm2);
19 \node[left] at (A) {$A$};
20 \node[right] at (B) {$B$};
21 \node[above] at (C) {$C$};
22 \node[below] at (O) {$O$};
23\end{tikzpicture}
24\caption{Proposition III.31. For any point $C$ on the circle (not at
25$A$ or $B$), the inscribed angle $\angle ACB$ subtending the diameter
26$AB$ is a right angle. Proof: by I.5 applied to the two isoceles
27triangles $OAC$ and $OCB$, then I.32.}
28\label{fig:III.31}
29\end{figure}
30