Thứ Bảy, 15 tháng 2, 2014

Bất ngờ từ tkz-euclide hay "với compa và thước kẻ thì có thể làm gì? Phần 4"

Ta có bài toán dựng hình:

Bài 18: Cho ba điểm A, B, C thẳng hàng theo thứ tự đó và ba nửa đường tròn đường kính AB, AC, BC nằm về một phía của đường thẳng AB. Dựng đường tròn tiếp xúc với cả ba nửa đường tròn trên.

Tiểu luận sinh viên Khoa Toán-Tin ĐHSP TP HCM năm 2010.


Giải bài toán dựng hình này là nội dung của môn Hình học sơ cấp và sử dụng phép nghịch đảo. Ở đây ta chỉ thảo luận về việc vẽ hình.


Bỏ qua giai đoạn phân tích bằng phép nghịch đảo, ta đưa ra vắn tắt cách dựng hình như sau:
  1. Dựng các nửa đường thẳng Bb; Cc lần lượt đi qua B, C và vuông góc với AC.
  2. Dựng đường tròn (L) tiếp xúc với nửa đường tròn đường kính BC và hai đường thẳng Bb; Cc, có tâm L cách trung điểm đoạn BC một khoảng bằng BC.
  3. Gọi (N) là đường tròn ảnh của (L) qua phép nghịch đảo cực A phương tích \(k=AB.AC\). (N) chính là đường tròn cần dựng.
Lưu ý: Không có công cụ là phép nghịch đảo thì dù phân tích được ta cũng không dễ dàng vẽ được đường tròn (N).

May thay tkz-euclide có công cụ phép nghịch đảo. Về điểm này, cho đến bây giờ tkz-euclide xứng đáng được công nhận là .

Code được đăng trên writelatex.com hoặc trên blog này:
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\documentclass[12pt]{article}
\usepackage[upright]{fourier} 
\usepackage{tkz-euclide}
\usetkzobj{all}
\usepackage[top=0cm,bottom=0cm,left=-.55cm,right=0cm,
paperwidth=12cm,paperheight=10cm]{geometry} 
\begin{document}
     
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-1,ymin=-1,xmax=11,ymax=7]
\tkzClip
\tkzDefPoint(0,0){A}
\tkzDefPoint(3,0){B}
\tkzDefPoint(7,0){C}

\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDrawLine[mediator,color=blue](A,C)

\tkzDefMidPoint(A,B)     
 \tkzGetPoint{I}
\tkzDrawArc[color=blue](I,B)(A)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)

\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}

\tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}

\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}

\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}

\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}

\tkzDrawCircle[circum,color=red,line width=1pt](U,K,D)
\end{tikzpicture}

\end{document}

 Giải thích: (xem flash minh họa ở dưới)

1. Các dòng

 
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\tkzDefPoint(0,0){A}
\tkzDefPoint(3,0){B}
\tkzDefPoint(7,0){C}

\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDrawLine[mediator,color=blue](A,C)

dùng để xác định các điểm A, B, C vẽ và gọi tên chúng. Sau đó vẽ đường thẳng AC.

2. Các dòng

 
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\tkzDefMidPoint(A,B)     
 \tkzGetPoint{I}
\tkzDrawArc[color=blue](I,B)(A)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

dùng để vẽ nửa đường tròn đường kính AB, BC, CA.

3. Các dòng

 
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\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)

\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)

dùng để vẽ các đường thẳng vuông góc với AC lần lượt tại B và C với ngọn là b và c.

4. Các dòng

 
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\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}

dùng để xác định đường tròn (L) có tâm L và bán kính bằng bán kính đường tròn đường kính BC. Điểm M dùng để xác định phương tích nghịch đảo.

Tkz-euclide định nghĩa phép nghịch đảo với cực là một điểm cố định và phương tích \(k\) là bình phương bán kính của đường tròn nghịch đảo (đường tròn có tâm là cực nghịch đảo và bán kính bằng \(\sqrt{k}\)). Trong bài này \(k=AB.AC\) và cụ thể bằng 21 nên \(\sqrt{21}=4.582575695\) là bán kính nói trên. Gọi M là một điểm trên đường tròn nghịch đảo. Phép nghịch đảo cực A phương tích \(k\), tkz-euclide gọi là phép nghịch đảo tâm A và đi qua M.

5. Các dòng 


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\tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}

\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}

\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}

\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}






Lấy ba điểm đặc biệt trên (L) là X(5,2); Y(7,4) , Z(3,4) chính là ba tiếp điểm của (L) với hai đường thẳng và đường tròn đường kính BC. Sau đó gọi ảnh của chúng qua phép nghịch đảo lần lượt  là U, K, D.



5. Cuối cùng






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\tkzDrawCircle[circum,color=red,line width=1pt](U,K,D)






vẽ đường tròn đi qua đi qua ba điểm ảnh ta được ảnh của (L) như hình vẽ.



Xem flash thầy Sơn thực hiện (khung ảnh lớn):

hoặc dưới đây:







Sau đây là code của flash


 \documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[vietnam]{babel}
\usepackage[upright]{fourier} 
\usepackage{tkz-euclide}
\usepackage{xcolor} 
\usetkzobj{all}
\usepackage[top=0cm,bottom=0cm,left=0cm,right=0cm,
paperwidth=12cm,paperheight=10cm]{geometry} 
\usepackage{multido}
\begin{document}
\newcommand{\debai}{{\large Cho ba điểm A, B, C thẳng hàng theo thứ tự đó và ba nửa đường tròn đường kính AB, AC, BC nằm về một phía của đường thẳng AB.  {\color{white} Dựng đường tròn tiếp xúc với cả ba nửa đường tròn trên.}}\vspace*{-2.5cm}

\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDefPoint(0,0){A}
\tkzDefPoint(3,0){B}
\tkzDefPoint(7,0){C}
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(A,B)     
\tkzGetPoint{I} 
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)


\tkzDefMidPoint(B,C)    
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)
\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)
\end{tikzpicture}}

\newcommand{\dapso}{{\large Cho ba điểm A, B, C thẳng hàng theo thứ tự đó và ba nửa đường tròn đường kính AB, AC, BC nằm về một phía của đường thẳng AB.  Dựng đường tròn tiếp xúc với cả ba nửa đường tròn trên.}\vspace*{-2.5cm}

\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)
\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)
\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)
\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDefPoint(5,4){L}
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}
\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}
\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}
\tkzDrawCircle[circum,color=red,line width=1pt](U,K,D)
\end{tikzpicture}}



\multido{}{10}{\newpage\debai}
\multido{}{5}{\newpage\dapso}
     
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDefPoint(0,0){A}
\tkzDefPoint(3,0){B}
\tkzDefPoint(7,0){C}

\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDrawLine[mediator,color=blue](A,C)
\end{tikzpicture}


\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDefPoint(0,0){A}
\tkzDefPoint(3,0){B}
\tkzDefPoint(7,0){C}

\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDrawLine[mediator,color=blue](A,C)
\end{tikzpicture}
\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDefPoint(0,0){A}
\tkzDefPoint(3,0){B}
\tkzDefPoint(7,0){C}

\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDrawLine[mediator,color=blue](A,C)
\tkzDefMidPoint(A,B)     
 \tkzGetPoint{I}
\tkzDrawArc[color=blue](I,B)(A)
\end{tikzpicture}
\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)
\end{tikzpicture}
\newpage

\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\end{tikzpicture}

\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\end{tikzpicture}

\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)
\end{tikzpicture}

\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
\end{tikzpicture}
\newpage

\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X)
\tkzDrawPoints(X)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}

\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}

\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}

\end{tikzpicture}

\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X)
\tkzDrawPoints(X)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}

\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}

\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}

\tkzDrawSegment[style=dashed,color=black](A,X)

\end{tikzpicture}
\newpage

\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X)
\tkzDrawPoints(X)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}


\tkzLabelPoints(U)
\tkzDrawPoints(U)
\tkzDrawSegment[style=dashed,color=black](A,X)
\end{tikzpicture}

\newpage

\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X,Y)
\tkzDrawPoints(X,Y)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}


\tkzLabelPoints(U)
\tkzDrawPoints(U)
\tkzDrawSegment[style=dashed,color=black](A,X)
\end{tikzpicture}




\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X,Y)
\tkzDrawPoints(X,Y)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}


\tkzLabelPoints(U)
\tkzDrawPoints(U)
\tkzDrawSegment[style=dashed,color=black](A,X)
\tkzDrawSegment[style=dashed,color=green](A,Y)
\end{tikzpicture}
\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X,Y)
\tkzDrawPoints(X,Y)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}

\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}

\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}

\tkzLabelPoints(U,K)
\tkzDrawPoints(U,K)
\tkzDrawSegment[style=dashed,color=green](A,Y)
\tkzDrawSegment[style=dashed,color=black](A,X)

\end{tikzpicture}
\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X,Y,Z)
\tkzDrawPoints(X,Y,Z)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}

\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}

\tkzGetPoint{D}

\tkzLabelPoints(U,K)
\tkzDrawPoints(U,K)
\tkzDrawSegment[style=dashed,color=green](A,Y)
\tkzDrawSegment[style=dashed,color=black](A,X)

\end{tikzpicture}

\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X,Y,Z)
\tkzDrawPoints(X,Y,Z)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}

\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}

\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}

\tkzLabelPoints(U,K)
\tkzDrawPoints(U,K)
\tkzDrawSegment[style=dashed,color=brown](A,Z)
\tkzDrawSegment[style=dashed,color=green](A,Y)
\tkzDrawSegment[style=dashed,color=black](A,X)

\end{tikzpicture}
\newpage
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X,Y,Z)
\tkzDrawPoints(X,Y,Z)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}

\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}

\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}

\tkzLabelPoints(U,K,D)
\tkzDrawPoints(U,K,D)
\tkzDrawSegment[style=dashed,color=brown](A,Z)
\tkzDrawSegment[style=dashed,color=green](A,Y)
\tkzDrawSegment[style=dashed,color=black](A,X)

\end{tikzpicture}

\newcommand{\hshc}{
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X,Y,Z)
\tkzDrawPoints(X,Y,Z)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}

\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}

\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}

\tkzLabelPoints(U,K,D)
\tkzDrawPoints(U,K,D)
\tkzDrawSegment[style=dashed,color=brown](A,Z)
\tkzDrawSegment[style=dashed,color=green](A,Y)
\tkzDrawSegment[style=dashed,color=black](A,X)
\end{tikzpicture}
}

\newcommand{\hchc}{
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)

\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)

\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)

\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDrawLine[end= $b$,add=0 and 1](B,b)
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDrawLine[end= $c$,add=0 and 0.5](C,c)


\tkzDefPoint(5,4){L}
\tkzDrawCircle[R,color=magenta,style=dashed](L, 2 cm)
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzLabelPoints(X,Y,Z)
\tkzDrawPoints(X,Y,Z)


\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}

\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}

\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}

\tkzLabelPoints(U,K,D)
\tkzDrawPoints(U,K,D)
\tkzDrawSegment[style=dashed,color=brown](A,Z)
\tkzDrawSegment[style=dashed,color=green](A,Y)
\tkzDrawSegment[style=dashed,color=black](A,X)

\tkzDrawCircle[circum,color=red,line width=1pt](U,K,D)
\end{tikzpicture}
}
\newcommand{\hht}{
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmin=-.5,ymin=-1,xmax=11.5,ymax=7]
\tkzClip
\tkzDrawLine[mediator,color=blue](A,C)
\tkzDrawArc[color=blue](I,B)(A)
\tkzDrawPoints(A,B,C)
\tkzLabelPoints(A,B,C)
\tkzDefMidPoint(B,C)     
\tkzGetPoint{J}
\tkzDrawArc[color=blue](J,C)(B)
\tkzDefMidPoint(A,C)     
\tkzGetPoint{K}
\tkzDrawArc[color=blue](K,C)(A)
\tkzDefLine[orthogonal=through B](A,B) \tkzGetPoint{b}
\tkzDefLine[orthogonal=through C](B,C) \tkzGetPoint{c}
\tkzDefPoint(5,4){L}
\tkzDefPoint(4.582575695,0){M}
 \tkzDefPoint(5,2){X}
\tkzDefPoint(7,4){Y}
\tkzDefPoint(3,4){Z}
\tkzDefPointBy[inversion = center A through M](X)
\tkzGetPoint{U}
\tkzDefPointBy[inversion = center A through M](Y)
\tkzGetPoint{K}
\tkzDefPointBy[inversion = center A through M](Z)
\tkzGetPoint{D}
\tkzDrawCircle[circum,color=red,line width=1pt](U,K,D)
\end{tikzpicture}
}

\multido{}{5}{\newpage\hshc}
\multido{}{10}{\newpage\hchc}
\multido{}{10}{\newpage\hht}
\end{document}










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