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 xếp đầu bảng.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
\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

12
13
14
15
16
17
18
19
\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

21
22
23
24
25
26
27
28
29
30
31
\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

33
34
35
36
37
\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

40
41
42
\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

\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

57
\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[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)  
 \end{tikzpicture}  
 \newpage  
 \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)  
 \end{tikzpicture}  
 \newpage  
 \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)  
 \end{tikzpicture}  
 \newpage  
 \begin{tikzpicture}[scale=1.25]  
 \tkzInit[xmin=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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=-1,ymin=-1,xmax=11,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}  
 \newpage  
 \begin{tikzpicture}[scale=1.25]  
 \tkzInit[xmin=-1,ymin=-1,xmax=11,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}  
 \newpage  
 \begin{tikzpicture}[scale=1.25]  
 \tkzInit[xmin=-1,ymin=-1,xmax=11,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}  
 \newpage  
 \begin{tikzpicture}[scale=1.25]  
 \tkzInit[xmin=-1,ymin=-1,xmax=11,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}  
 \newpage  
 \begin{tikzpicture}[scale=1.25]  
 \tkzInit[xmin=-1,ymin=-1,xmax=11,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}\newpage  
 \begin{tikzpicture}[scale=1.25]  
 \tkzInit[xmin=-1,ymin=-1,xmax=11,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}\newpage  
 \begin{tikzpicture}[scale=1.25]  
 \tkzInit[xmin=-1,ymin=-1,xmax=11,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}\newpage  
 \begin{tikzpicture}[scale=1.25]  
 \tkzInit[xmin=-1,ymin=-1,xmax=11,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}\newpage  
 \begin{tikzpicture}[scale=1.25]  
 \tkzInit[xmin=-1,ymin=-1,xmax=11,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}  
 \end{document}  

Gửi phản hồi

Mời bạn điền thông tin vào ô dưới đây hoặc kích vào một biểu tượng để đăng nhập:

WordPress.com Logo

Bạn đang bình luận bằng tài khoản WordPress.com Log Out / Thay đổi )

Twitter picture

Bạn đang bình luận bằng tài khoản Twitter Log Out / Thay đổi )

Facebook photo

Bạn đang bình luận bằng tài khoản Facebook Log Out / Thay đổi )

Google+ photo

Bạn đang bình luận bằng tài khoản Google+ Log Out / Thay đổi )

Connecting to %s