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绘制三角形三边的垂直平分线

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绘制三角形三边的垂直平分线

在此处输入图片描述

（1） ，L和M必分别为边，和N的中点。ABACBC

（2） ，LX和MY必NZ为垂线。

（3）注意LX和NZ必定在点相交P，并且MY必定在通过点之前终止P。

 \documentclass[11pt,a4paper]{article}
 \usepackage{blindtext}
 \usepackage{tikz}
 \usepackage{tkz-euclide}
 \usetkzobj{all}
 \usepackage{color}

 \begin{document}
 \normalsize{\textbf{Theorem 1.25.} \textit{The three perpendicular bisectors of the sides of a triangle meet in a point which is equally distant from the vertices.}}
 \begin{center}
\begin{tikzpicture}
\clip
(-1,-3) rectangle (13,8);
\tkzDefPoint(0,0){A}
\tkzDefPoint(12,0){B}
\tkzLabelPoints[below](A)
\tkzLabelPoints[below](B)
\tkzDrawSegment(A,B)
\tkzDefPoint(4,7){C}
\tkzLabelPoints[above](C)
\tkzDrawSegment(A,C)
\tkzDrawSegment(B,C)


\end{tikzpicture}
\end{center}
 \end{document}

答案1

要找到中点L，M，N：

\tkzDefMidPoint(A,B) \tkzGetPoint{L}

寻找线段平分线上的辅助点X，Y， ：Z

\tkzDefLine[orthogonal=through L](A,B) \tkzGetPoint{X}

或者

\tkzDefLine[mediator](A,B) \tkzGetPoint{X}

求P两个角平分线的交点AX和BY：

\tkzInterLL(L,X)(M,Y) \tkzGetPoint{P}


 \documentclass[11pt,a4paper]{article}
 \usepackage{blindtext}
 \usepackage{tikz}
 \usepackage{tkz-euclide}
 \usetkzobj{all}
 \usepackage{color}

 \begin{document}
 \normalsize{\textbf{Theorem 1.25.} \textit{The three perpendicular bisectors of the sides of a triangle meet in a point which is equally distant from the vertices.}}
 \begin{center}
\begin{tikzpicture}
\clip
(-1,-3) rectangle (13,8);
\tkzDefPoint(0,0){A}
\tkzDefPoint(12,0){B}
\tkzDefPoint(4,7){C}

\tkzDrawSegment(A,B)
\tkzDrawSegment(B,C)
\tkzDrawSegment(C,A)

\tkzDefMidPoint(A,B) \tkzGetPoint{L}
\tkzDefLine[orthogonal=through L](A,B) \tkzGetPoint{X}

\tkzDefMidPoint(B,C) \tkzGetPoint{M}
\tkzDefLine[orthogonal=through M](B,C) \tkzGetPoint{Y}

\tkzDefMidPoint(C,A) \tkzGetPoint{N}

\tkzInterLL(L,X)(M,Y) \tkzGetPoint{P}

\tkzDrawLines(L,X M,Y)
\tkzDrawSegment(N,P)
\tkzMarkRightAngle(A,L,X)
\tkzMarkRightAngle(B,M,Y)
\tkzMarkRightAngle(C,N,P)

\tkzLabelPoints(B,L)
\tkzLabelPoints[below left](A)
\tkzLabelPoints[above](C,M)
\tkzLabelPoints[above left](N)
\tkzLabelPoints[below right](P)

\end{tikzpicture}
\end{center}
 \end{document}

答案2

使用 MetaPost，作为可能感兴趣的人的补充。在MetaPost 手册和André Heck 的 MetaPost 教程对于宏mark_right_angle，draw_mark和tick。

找到中点：

L = .5[A, B]; M = .5[A, C]; N = .5[B, C];

交点是通过隐式方程找到的，就像 MetaPost 中经常使用的那样：

P = whatever[L, L + (B-A) rotated 90] = whatever[N, N + (C-B) rotated 90];

X、Y 和 Z 是在中点和数值参数的帮助下构建的over（under相对于 P 的位置）。

X = P + over*unitvector(P-L); 
Y = P - under*unitvector(P-M); 
Z = P + over*unitvector(P-N);

整个代码：

\documentclass{scrartcl}
\usepackage{luamplib}
    \mplibtextextlabel{enable}
\begin{document}
\begin{mplibcode}

vardef mark_right_angle (expr common, endofa, endofb, size) = % right angle mark
    save tn ; tn := turningnumber(common -- endofa -- endofb -- cycle) ;
    draw ((1, 0) -- (1, 1) -- (0, 1))
        zscaled (size*unitvector((1+tn)*endofa + (1-tn)*endofb - 2*common))
        shifted common;
enddef ;

vardef draw_mark(expr p, m, size) = % One mark upon a segment
    save t, dm; pair dm;
    t = arctime m of p;
    dm = size*unitvector(direction t of p rotated 90);
    draw (-.5dm .. .5dm) shifted (point t of p); 
enddef;


vardef tick(expr p, n, size) = % Several marks upon a segment
    save midpnt; midpnt = 0.5*arclength(p);
    for i = -(n-1)/2 upto (n-1)/2:
        draw_mark(p, midpnt+0.6size*i/2, size); 
    endfor;
enddef;

u := 1cm; over := u; under := 0.75u;
pair A, B, C, L, M, N, P, X, Y, Z; path triangle;
A = origin; B = (12u, 0); C = u*(4, 7);
triangle = A -- B -- C -- cycle; 
L = .5[A, B]; M = .5[A, C]; N = .5[B, C];
% Locating the intersection
P = whatever[L, L + (B-A) rotated 90] = whatever[N, N + (C-B) rotated 90];
% Bisectors 
X = P + over*unitvector(P-L); 
Y = P - under*unitvector(P-M); 
Z = P + over*unitvector(P-N);

beginfig(1);
    rsize := 2mm; msize := 3mm;
    draw triangle; draw L -- X; draw M -- Y; draw N -- Z;
    mark_right_angle(L, B, P, rsize); tick(A--L, 2, msize); tick(L--B, 2, msize);
    mark_right_angle(M, C, P, rsize); tick(A--M, 1, msize); tick(C--M, 1, msize); 
    mark_right_angle(N, C, P, rsize); tick(C--N, 3, msize); tick(B--N, 3, msize);
    label.llft("$A$", A); label.bot("$B$", B);
    label.top("$C$", C); label.rt("$P$", P);
    label.llft("$L$", L); label.ulft("$M$", M);
    label.urt("$N$", N); label.top("$X$", X);
    label.bot("$Y$", Y); label.lft("$Z$", Z);
endfig;

\end{mplibcode}
\end{document}

要用 LuaLaTeX 处理：

在此处输入图片描述

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