![](https://i0.wp.com/35.221.254.218/wp-content/uploads/2023/05/right_triangle_007.png?resize=379%2C415)
\begin{equation}延申\mathrm{AB}到\mathrm{H},並作直線\mathrm{DE}\perp\mathrm{AH},則\end{equation}\begin{equation}\triangle\mathrm{ABC}\sim\triangle\mathrm{AEH}\end{equation}\begin{equation}\triangle\mathrm{ABC}\sim\triangle\mathrm{DEC}\end{equation}\begin{equation}\triangle\mathrm{ABC}\sim\triangle\mathrm{DBH}\end{equation}
\begin{equation}讓\mathrm{AB}=\mathrm{c},\mathrm{BC}=\mathrm{a},\mathrm{AC}=\mathrm{b},\mathrm{BD}=\mathrm{x},\end{equation}\begin{equation}\mathrm{CE}=\mathrm{y},\mathrm{BH}=\mathrm{z}。\end{equation}
\begin{equation}\mathrm{1}. \triangle\mathrm{ABC}\sim\triangle\mathrm{AEH}\end{equation}\begin{equation}\therefore\mathrm{AB}:\mathrm{AE}=\mathrm{AC}:\mathrm{AH}\end{equation}\begin{equation}\Rightarrow\mathrm{c}:(\mathrm{b}+\mathrm{y})=\mathrm{b}:(\mathrm{c}+\mathrm{z})\end{equation}\begin{equation}\Rightarrow\mathrm{b}^2+\mathrm{by}=\mathrm{c}^2+\mathrm{cz}\cdots\cdots(\mathrm{1})\end{equation}
\begin{equation}\mathrm{2}. \triangle\mathrm{ABC}\sim\triangle\mathrm{DEC}\end{equation}\begin{equation}\therefore\mathrm{BC}:\mathrm{EC}=\mathrm{AC}:\mathrm{DC}\end{equation}\begin{equation}\Rightarrow\mathrm{a}:\mathrm{y}=\mathrm{b}:(\mathrm{a}+\mathrm{x})\end{equation}\begin{equation}\Rightarrow\mathrm{by}=\mathrm{a}^2+\mathrm{ax}\cdots\cdots(\mathrm{2})\end{equation}
\begin{equation}\mathrm{3}. \triangle\mathrm{ABC}\sim\triangle\mathrm{DBH}\end{equation}\begin{equation}\therefore\mathrm{AB}:\mathrm{DB}=\mathrm{BC}:\mathrm{BH}\end{equation}\begin{equation}\Rightarrow\mathrm{c}:\mathrm{x}=\mathrm{a}:\mathrm{z}\end{equation}\begin{equation}\Rightarrow\mathrm{ax}=\mathrm{cz}\cdots\cdots(\mathrm{3})\end{equation}
\begin{equation}將(\mathrm{3})(\mathrm{2})代入(\mathrm{1})可以即可得到\end{equation}
\begin{equation}\Rightarrow\mathrm{a}^2+\mathrm{b}^2=\mathrm{c}^2\end{equation}
Reference:
1. Elisha S. Loomis (1935): « The Pythagorean Proposition ». p. 28.