% fig-ii-4.tex — II.4: binomial square (a+b)^2 = a^2 + 2ab + b^2.
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=1.0, line cap=round, line join=round]
  \def\a{2.4}
  \def\b{1.4}
  \coordinate (A) at (0, 0);
  \coordinate (B) at ({\a+\b}, 0);
  \coordinate (D) at (0, {\a+\b});
  \coordinate (E) at ({\a+\b}, {\a+\b});
  \coordinate (C) at (\a, 0);
  \coordinate (G) at (\a, \a);
  \coordinate (F) at (\a, {\a+\b});
  \coordinate (H) at (0, \a);
  \coordinate (K) at ({\a+\b}, \a);
  % Outer square ADEB.
  \draw[very thick] (A) -- (B) -- (E) -- (D) -- cycle;
  % Diagonal BD.
  \draw[thin, dashed] (B) -- (D);
  % Internal partition: CF parallel to AD, HK parallel to AB.
  \draw[thick] (C) -- (F);
  \draw[thick] (H) -- (K);
  % Shaded squares: HDFG (top-left, side a) and CBKG (bottom-right, side b).
  \fill[gray!10] (H) -- (D) -- (F) -- (G) -- cycle;
  \fill[gray!25] (C) -- (B) -- (K) -- (G) -- cycle;
  % Re-draw lines on top so shading doesn't hide them.
  \draw[very thick] (A) -- (B) -- (E) -- (D) -- cycle;
  \draw[thick] (C) -- (F);
  \draw[thick] (H) -- (K);
  \draw[thin, dashed] (B) -- (D);
  % Labels.
  \node[below left]  at (A) {$A$};
  \node[below]       at (C) {$C$};
  \node[below right] at (B) {$B$};
  \node[above left]  at (D) {$D$};
  \node[above]       at (F) {$F$};
  \node[above right] at (E) {$E$};
  \node[left]        at (H) {$H$};
  \node[right]       at (K) {$K$};
  \node              at (G) {$G$};
  % Side annotations.
  \node[below] at ($(A)!0.5!(C)$) {$a$};
  \node[below] at ($(C)!0.5!(B)$) {$b$};
\end{tikzpicture}
\caption{Proposition II.4. The square $ADEB$ on $AB = a+b$ is
decomposed by the parallels $CF \parallel AD$ and $HK \parallel AB$
into the square $HDFG$ on $AC = a$, the square $CBKG$ on $CB = b$, and
two equal rectangles $AGHD$ and $GFBK$ (by I.43), each equal to
$a \cdot b$.}
\label{fig:II.4}
\end{figure}
