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1% fig-ii-11.tex — II.11: cut a line in extreme and mean ratio (golden section).
2\begin{figure}[H]
3\centering
4\begin{tikzpicture}[scale=1.2, line cap=round, line join=round]
5 \coordinate (A) at (0, 0);
6 \coordinate (B) at (3, 0);
7 \coordinate (C) at (0, 3);
8 \coordinate (D) at (3, 3);
9 % Square ABDC on AB.
10 \draw[very thick] (A) -- (B) -- (D) -- (C) -- cycle;
11 % E = midpoint of AC.
12 \coordinate (E) at (0, 1.5);
13 \draw[thin, dashed] (E) -- (B);
14 % F = on AC produced, with EF = EB. EF = sqrt(9 + 2.25) ~ 3.354.
15 \coordinate (F) at (0, {-sqrt(11.25) + 1.5});
16 \draw[thin] (E) -- (F);
17 % H on AB with AH = AF.
18 \coordinate (H) at ({sqrt(11.25) - 1.5}, 0);
19 \draw[very thick, dotted] (H) -- ($(H)+(0, 3)$);
20 % Labels.
21 \node[above left] at (A) {$A$};
22 \node[above right] at (B) {$B$};
23 \node[below right] at (D) {$D$};
24 \node[below left] at (C) {$C$};
25 \node[left] at (E) {$E$};
26 \node[left] at (F) {$F$};
27 \node[below] at (H) {$H$};
28 % Arc from F to H (centre A, radius AF).
29 \draw[thin] (F) arc[start angle=270, end angle=360, radius={sqrt(11.25) - 1.5}];
30\end{tikzpicture}
31\caption{Proposition II.11. Square $ABDC$ on $AB$; midpoint $E$ of $AC$;
32$F$ on the extension of $AC$ with $EF = EB$. Then $AH = AF$ cuts $AB$
33in the desired ratio: $AB \cdot HB = AH^2$. This is the golden section.}
34\label{fig:II.11}
35\end{figure}
36