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Figure
1% fig-i-47.tex — I.47: Pythagoras (square decomposition).
2\begin{figure}[H]
3\centering
4\begin{tikzpicture}[scale=0.55, line cap=round, line join=round]
5 % Right triangle vertices: right angle at A.
6 \coordinate (A) at (0, 0);
7 \coordinate (B) at (3, 0); % horizontal leg
8 \coordinate (C) at (0, 4); % vertical leg
9 % Hypotenuse BC.
10 % Square on AB (below): A, B, B', A'
11 \coordinate (Ap) at (0, -3);
12 \coordinate (Bp) at (3, -3);
13 % Square on AC (left): A, C, C', A''
14 \coordinate (App) at (-4, 0);
15 \coordinate (Cp) at (-4, 4);
16 % Square on BC (outward): B, C, C'', B''
17 % Outward direction = rotate (B-C) by -90.
18 \coordinate (Bpp) at ($(B)!1!-90:(C)$);
19 \coordinate (Cpp) at ($(C)!1!90:(B)$);
20 % Triangle.
21 \draw[very thick] (A) -- (B) -- (C) -- cycle;
22 % Three squares.
23 \draw[thick, fill=gray!10] (A) -- (B) -- (Bp) -- (Ap) -- cycle;
24 \draw[thick, fill=gray!10] (A) -- (C) -- (Cp) -- (App) -- cycle;
25 \draw[thick, fill=gray!20] (B) -- (C) -- (Cpp) -- (Bpp) -- cycle;
26 % Altitude from A to BC, foot at H, extended to meet square on BC.
27 \coordinate (H) at ($(B)!(A)!(C)$);
28 \coordinate (Hext) at ($(H)!1!-90:(B)$);
29 \draw[thin, dashed] (A) -- (Hext);
30 % Labels.
31 \node[below right] at (A) {$A$};
32 \node[below] at (B) {$B$};
33 \node[left] at (C) {$C$};
34 \node[left] at ($(A)!0.5!(App)$) {square on $AC$};
35 \node at ($(A)!0.5!(Bp)+(0.5,-1.5)$) {square on $AB$};
36 \node at ($(B)!0.5!(Cpp)+(1.7,0.7)$) {square on $BC$};
37\end{tikzpicture}
38\caption{Proposition I.47. The square on the hypotenuse $BC$ is
39partitioned by the altitude $AH$ extended into two rectangles, each
40equal (by I.41 + I.46) to a square on a leg; thus $BC^2 = AB^2 + AC^2$.}
41\label{fig:I.47}
42\end{figure}
43