做法

作圆的垂直中心线AB,CDAB,\,CDAB,CD;
以BBB为圆心, BCBCBC为半径作圆弧与ABABAB交于点GGG;
以GGG为圆心, GCGCGC为半径作圆弧交ABABAB于FFF点;
以BBB为圆心, BFBFBF为半径作圆弧交圆周于EEE点;
连接AEAEAE交圆弧CF^\widehat{CF}CF于HHH点;
连接OHOHOH, OHOHOH即为内接正九边形的边长.

图形

代码

\documentclass[tikz,border=3pt]{standalone}
\usetikzlibrary{calc}
\usetikzlibrary{intersections,through}\begin{document}\begin{tikzpicture}[]% 标记点A,B,C,D,O\coordinate [label=below left:$A$] (A) at (0,0);\coordinate [label=below right:$B$] (B) at (4,0);\coordinate [label=above left:$C$] (C) at (2,2);\coordinate [label=below right:$D$] (D) at (2,-2);\coordinate [label=below left:$O$] (O) at (2,0);% 圆o\draw [name path=o] (O) circle (2);% AB, CD为圆的垂直中心线\draw [densely dashed,-latex,name path=AB] ($(A)!-.1!(B)$) -- ($(A)!1.1!(B)$);\draw [densely dashed,latex-,name path=CD] ($(C)!-.1!(D)$) -- ($(C)!1.1!(D)$);% 以B为圆心, BC为半径\path [name path=oCG]let \p1=($(B)-(C)$),\n1={veclen(\x1,\y1)}in (B) circle (\n1);% 标记圆弧与AB的交点G\path [name intersections={of=oCG and AB}]coordinate [label=below left:$G$] (G) at (intersection-1);% 绘制弧段\draw [red]let \p1=($(B)-(C)$),\n1={veclen(\x1,\y1)}in (G) arc (180:120:\n1)(G) arc (180:185:\n1);% 以G为圆心, GC为半径\path [name path=oCF]let \p1=($(G)-(C)$),\n1={veclen(\x1,\y1)}in (G) circle (\n1);\path [name intersections={of=AB and oCF}]coordinate [label=below left:$F$] (F) at (intersection-1);\draw [red] let \p1=($(G)-(C)$),\n1={veclen(\x1,\y1)}in (F) arc (0:75:\n1)(F) arc (0:-5:\n1);% 以B为圆心, BF为半径\path [name path=oEF]let \p1=($(B)-(F)$),\n1={veclen(\x1,\y1)}in (B) circle (\n1);\path [name intersections={of=o and oEF}]coordinate [label=right:$E$] (E) at (intersection-1);\draw [red] let \p1=($(B)-(F)$),\n1={veclen(\x1,\y1)}in (F) arc (180:90:\n1)(F) arc (180:190:\n1);% 连接AE交圆弧CF\draw [name path=AE] (A) -- (E);\path [name intersections={of=oCF and AE}]coordinate [label=above right:$H$] (H) at (intersection-1);% 连接OH\draw [thick] (O) -- (H);% 找到其他点\path [name path=o2]let\p2=($ (O) - (H) $),\n2={veclen(\x2,\y2)}in (C) circle (\n2);\path [name intersections={of=o2 and o}]coordinate [] (C1) at (intersection-1)coordinate [] (C8) at (intersection-2);\path [name path=o3]let\p2=($ (O) - (H) $),\n2={veclen(\x2,\y2)}in (C1) circle (\n2);\path [name intersections={of=o3 and o}]coordinate [] (C2) at (intersection-2);\path [name path=o4]let\p2=($ (O) - (H) $),\n2={veclen(\x2,\y2)}in (C2) circle (\n2);\path [name intersections={of=o4 and o}]coordinate [] (C3) at (intersection-2);\path [name path=o5]let\p2=($ (O) - (H) $),\n2={veclen(\x2,\y2)}in (C3) circle (\n2);\path [name intersections={of=o5 and o}]coordinate [] (C4) at (intersection-2);\path [name path=o6]let\p2=($ (O) - (H) $),\n2={veclen(\x2,\y2)}in (C4) circle (\n2);\path [name intersections={of=o6 and o}]coordinate [] (C5) at (intersection-1);\path [name path=o7]let\p2=($ (O) - (H) $),\n2={veclen(\x2,\y2)}in (C5) circle (\n2);\path [name intersections={of=o7 and o}]coordinate [] (C6) at (intersection-1);\path [name path=o8]let\p2=($ (O) - (H) $),\n2={veclen(\x2,\y2)}in (C6) circle (\n2);\path [name intersections={of=o8 and o}]coordinate [] (C7) at (intersection-1);% 绘制正九边形\draw [thick] (C) -- (C1) -- (C2) -- (C3) -- (C4) -- (C5) -- (C6) -- (C7) -- (C8) -- cycle;\end{tikzpicture}
\end{document}

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TikZ绘图示例——尺规作图: 圆内接正九边形的近似画法

做法

图形

代码

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