## Example 1 Gou gu theorem / Pythagorean Theorem by Zhao Shuang

Pythagoras was not the first person who discovered this theorem around the world. Ancient China discovered this theorem much earlier than him. So there is another name for the Pythagorean theorem in China, the Gou-Gu theorem. Zhao Shuang was an ancient Chinese mathematician. He rediscovered the “Gou gu therorem”, which is actually the Chinese version of the “Pythagorean theorem”. Zhao Shuang used a method called the “cutting and compensation principle”, he created a picture of “Pythagorean Round Square” Below the figure used to illustrate the proof of the “Gou gu theorem.” (code from Nan Geng).

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```\documentclass{standalone}
\usepackage{tkz-euclide}
\tkzSetUpPoint[size=3,color=teal,fill=teal!10]
\tkzSetUpLine[line width=0.4pt,color=teal]
\tkzSetUpCompass[color=red,ultra thick,delta=0]
\begin{document}
\begin{tikzpicture}[scale=.8]
\tkzDefPoint(0,0){A} \tkzDefPoint(4,0){A'}
\tkzInterCC[R](A, 5)(A', 3)
\tkzGetSecondPoint{B}
\tkzDefSquare(A,B)   \tkzGetPoints{C}{D}
\tkzCalcLength(A,A') \tkzGetLength{lA}
\tkzCalcLength(A',B) \tkzGetLength{lB}
\pgfmathparse{\lA-\lB}
\tkzInterLC[R](A,A')(A',\pgfmathresult)
\tkzGetFirstPoint{D'}
\tkzDefSquare(D',A')\tkzGetPoints{B'}{C'}
\tkzDefLine[orthogonal=through D](D,D')
\tkzGetPoint{d}
\tkzDefLine[orthogonal=through A](A,A')
\tkzGetPoint{a}
\tkzDefLine[orthogonal=through C](C,C')
\tkzGetPoint{c}
\tkzInterLL(D,d)(C,c) \tkzGetPoint{E}
\tkzInterLL(D,d)(A,a) \tkzGetPoint{F}
\tkzDefSquare(E,F)\tkzGetPoints{G}{H}
\tkzDrawPolygons[fill=teal!10](A,B,A' B,C,B' C,D,C' A,D',D)
\tkzDrawPolygons(A,B,C,D E,F,G,H)
\tkzDrawPolygon[fill=green!10](A',B',C',D')
\tkzDrawSegment[dim={\$a\$,-10pt,}](D,C')
\tkzDrawSegment[dim={\$b\$,-10pt,}](C,C')
\tkzDrawSegment[dim={\$c\$,-10pt,}](C,D)
\tkzDrawPoints[size=2](A,B,C,D,A',B',C',D')
\tkzLabelPoints[left](A)
\tkzLabelPoints[below](B)
\tkzLabelPoints[right](C)
\tkzLabelPoints[above](D)
\tkzLabelPoints[right](A')
\tkzLabelPoints[below right](B')
\tkzLabelPoints[below left](C')
\tkzLabelPoints[below](D')
\end{tikzpicture}
\end{document}```