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1% fig-ii-4.tex — II.4: binomial square (a+b)^2 = a^2 + 2ab + b^2.
2\begin{figure}[H]
3\centering
4\begin{tikzpicture}[scale=1.0, line cap=round, line join=round]
5 \def\a{2.4}
6 \def\b{1.4}
7 \coordinate (A) at (0, 0);
8 \coordinate (B) at ({\a+\b}, 0);
9 \coordinate (D) at (0, {\a+\b});
10 \coordinate (E) at ({\a+\b}, {\a+\b});
11 \coordinate (C) at (\a, 0);
12 \coordinate (G) at (\a, \a);
13 \coordinate (F) at (\a, {\a+\b});
14 \coordinate (H) at (0, \a);
15 \coordinate (K) at ({\a+\b}, \a);
16 % Outer square ADEB.
17 \draw[very thick] (A) -- (B) -- (E) -- (D) -- cycle;
18 % Diagonal BD.
19 \draw[thin, dashed] (B) -- (D);
20 % Internal partition: CF parallel to AD, HK parallel to AB.
21 \draw[thick] (C) -- (F);
22 \draw[thick] (H) -- (K);
23 % Shaded squares: HDFG (top-left, side a) and CBKG (bottom-right, side b).
24 \fill[gray!10] (H) -- (D) -- (F) -- (G) -- cycle;
25 \fill[gray!25] (C) -- (B) -- (K) -- (G) -- cycle;
26 % Re-draw lines on top so shading doesn't hide them.
27 \draw[very thick] (A) -- (B) -- (E) -- (D) -- cycle;
28 \draw[thick] (C) -- (F);
29 \draw[thick] (H) -- (K);
30 \draw[thin, dashed] (B) -- (D);
31 % Labels.
32 \node[below left] at (A) {$A$};
33 \node[below] at (C) {$C$};
34 \node[below right] at (B) {$B$};
35 \node[above left] at (D) {$D$};
36 \node[above] at (F) {$F$};
37 \node[above right] at (E) {$E$};
38 \node[left] at (H) {$H$};
39 \node[right] at (K) {$K$};
40 \node at (G) {$G$};
41 % Side annotations.
42 \node[below] at ($(A)!0.5!(C)$) {$a$};
43 \node[below] at ($(C)!0.5!(B)$) {$b$};
44\end{tikzpicture}
45\caption{Proposition II.4. The square $ADEB$ on $AB = a+b$ is
46decomposed by the parallels $CF \parallel AD$ and $HK \parallel AB$
47into the square $HDFG$ on $AC = a$, the square $CBKG$ on $CB = b$, and
48two equal rectangles $AGHD$ and $GFBK$ (by I.43), each equal to
49$a \cdot b$.}
50\label{fig:II.4}
51\end{figure}
52