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Fax: \ 41 22 / 346 59 39", FontSize->18] }], "Text", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Introduction", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ In the seventeenth century, two french mathematicians, Pierre de Fermat and \ Ren\[CapitalIHat] Descartes, realized that algebra and geometry could be \ unified in a single science that was called analytic geometry. In fact, the \ recognition for the foundation of the new science went to Descartes although \ Fermat's work was a little earlier. The reason is due to the formalism used \ by Descartes who adopted the notation of modern algebra whereas Fermat had \ kept the cumbersome greek notations. Today, we can resume the treatment of \ analytic geometry in the context of the modern computers and information \ languages among which Mathematica is especially well adapted with its \ outstanding graphics facilities and its functional programming. \t\t\t\tLet us start\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Constructions", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ In this chapter, we illustrate a few classical constructions.\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Tangents", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The symbol ", StyleBox["Dot ", FontWeight->"Bold"], "(", StyleBox[".", FontWeight->"Bold"], ")", StyleBox[" ", FontWeight->"Bold"], "has different meanings according to the nature of the arguments. When the \ arguments are a point and a circle, it returns the tangents to the circle \ issued from the point." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ tg=c.c1 Draw[Blue,PointSize[.02],{a,\"A\"},{c,\"C\"},c1, Red,tg,Interval->{{-1,2},{-1,2}}];\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {STRAIGHT[1/3 + (2*5^(1/2))/3, 2^(1/2) - 10^(1/2)/9, -19/9], STRAIGHT[-1/3 + (2*5^(1/2))/3, -2^(1/2) - 10^(1/2)/9, 19/9]}\ \>", "\<\ 1 2 Sqrt[5] Sqrt[10] 19 {STRAIGHT[- + ---------, Sqrt[2] - --------, -(--)], 3 3 9 9 1 2 Sqrt[5] Sqrt[10] 19 STRAIGHT[-(-) + ---------, -Sqrt[2] - --------, --]} 3 3 9 9\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.09486 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.371385 0.347576 0.373644 0.347576 [ [(A)] 0.37139 0.37364 -1 -1 Msboxa [(C)] 0.48724 0.86519 -1 -1 Msboxa [ 0 0 0 0 ] [ 1 1.09486 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore gsave 0 0 1 setrgbcolor gsave 0.02 setlinewidth 0.37139 0.37364 Mdot [(A)] 0.37139 0.37364 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.48724 0.86519 Mdot [(C)] 0.48724 0.86519 -1 -1 Mshowa grestore 0.004 setlinewidth newpath 0.37139 0.37364 0.34758 0 365.73 arc stroke 1 0 0 setrgbcolor gsave 0.36861 1.0688 moveto 0.97619 0.02607 lineto stroke 0.02381 0.5614 moveto 0.79784 1.0688 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 1.09486 lineto 0 1.09486 lineto closepath clip newpath % End of Graphics MathPictureEnd\ \>"], "Graphics", ImageSize->{281, 307}, ImageMargins->{{34, Inherited}, {Inherited, Inherited}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Intersections", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The symbol ", StyleBox["Times ", FontWeight->"Bold"], "or", StyleBox[" *", FontWeight->"Bold"], " defines any intersection. We use it to find the contact points of the \ tangents we have just found with the circle." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ it=(tg*c1)//Flatten Draw[Blue,PointSize[.02],{a,\"A\"},{c,\"C\"},c1,tg, Red,{it[[1]],\"it1\"},{it[[2]],\"it2\",{1,-1}}, Interval->{{-1,2},{-1,2}}];\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {POINT[(19*(1/3 + (2*5^(1/2))/3))/ (9*((1/3 + (2*5^(1/2))/3)^2 + (2^(1/2) - 10^(1/2)/9)^2)), (19*(2^(1/2) - 10^(1/2)/9))/ (9*((1/3 + (2*5^(1/2))/3)^2 + (2^(1/2) - 10^(1/2)/9)^2))], POINT[(-19*(-1/3 + (2*5^(1/2))/3))/ (9*((-1/3 + (2*5^(1/2))/3)^2 + (-2^(1/2) - 10^(1/2)/9)^2)), (-19*(-2^(1/2) - 10^(1/2)/9))/ (9*((-1/3 + (2*5^(1/2))/3)^2 + (-2^(1/2) - 10^(1/2)/9)^2))]}\ \>", "\<\ 1 2 Sqrt[5] 19 (- + ---------) 3 3 {POINT[--------------------------------------------, 1 2 Sqrt[5] 2 Sqrt[10] 2 9 ((- + ---------) + (Sqrt[2] - --------) ) 3 3 9 Sqrt[10] 19 (Sqrt[2] - --------) 9 --------------------------------------------], 1 2 Sqrt[5] 2 Sqrt[10] 2 9 ((- + ---------) + (Sqrt[2] - --------) ) 3 3 9 POINT[ 1 2 Sqrt[5] -19 (-(-) + ---------) 3 3 ------------------------------------------------, 1 2 Sqrt[5] 2 Sqrt[10] 2 9 ((-(-) + ---------) + (-Sqrt[2] - --------) ) 3 3 9 Sqrt[10] -19 (-Sqrt[2] - --------) 9 ------------------------------------------------]} 1 2 Sqrt[5] 2 Sqrt[10] 2 9 ((-(-) + ---------) + (-Sqrt[2] - --------) ) 3 3 9\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.09486 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.371385 0.347576 0.373644 0.347576 [ [(A)] 0.37139 0.37364 -1 -1 Msboxa [(C)] 0.48724 0.86519 -1 -1 Msboxa [(it1)] 0.6717 0.54863 -1 -1 Msboxa [(it2)] 0.18083 0.66433 1 -1 Msboxa [ 0 0 0 0 ] [ 1 1.09486 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore gsave 0 0 1 setrgbcolor gsave 0.02 setlinewidth 0.37139 0.37364 Mdot [(A)] 0.37139 0.37364 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.48724 0.86519 Mdot [(C)] 0.48724 0.86519 -1 -1 Mshowa grestore 0.004 setlinewidth newpath 0.37139 0.37364 0.34758 0 365.73 arc stroke gsave 0.36861 1.0688 moveto 0.97619 0.02607 lineto stroke 0.02381 0.5614 moveto 0.79784 1.0688 lineto stroke grestore 1 0 0 setrgbcolor gsave 0.02 setlinewidth 0.6717 0.54863 Mdot [(it1)] 0.6717 0.54863 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.18083 0.66433 Mdot [(it2)] 0.18083 0.66433 1 -1 Mshowa grestore grestore 0 0 moveto 1 0 lineto 1 1.09486 lineto 0 1.09486 lineto closepath clip newpath % End of Graphics MathPictureEnd\ \>"], "Graphics", ImageSize->{277, 303}, ImageMargins->{{34, Inherited}, {Inherited, Inherited}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Perpendicular Bisector", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The perpendicular bisector of a segment ", StyleBox["ab", FontWeight->"Bold"], " is the straight line which cuts the segment in its middle and at right \ angle. It is noted\n a \ ", StyleBox["+ ", FontWeight->"Bold"], "b\nWe determine the perpendi cular bisector of the segment which joins the \ contact points ", StyleBox["it", FontWeight->"Bold"], ". 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The remark about the operator ", StyleBox["Plus", FontWeight->"Bold"], " in the previous section is valid here for the operator ", StyleBox["Dot", FontWeight->"Bold"], "." }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ d5=Dot@@it Draw[Blue,PointSize[.02],{a,\"A\"},c1, {it[[1]],\"it1\"},{it[[2]],\"it2\",{1,-1}}, Red,d5,PointSize[0],{POINT[1.7,0.3],\"d5\"}, Interval->{{-1,2},{-1,2}}];\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ STRAIGHT[-40^(1/2)/19, (-12*5^(1/2))/19, (3*40^(1/2))/19]\ \>", "\<\ -Sqrt[40] -12 Sqrt[5] 3 Sqrt[40] STRAIGHT[---------, -----------, ----------] 19 19 19\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 0.66667 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.34127 0.31746 0.333333 0.31746 [ [(A)] 0.34127 0.33333 -1 -1 Msboxa [(it1)] 0.61556 0.49316 -1 -1 Msboxa [(it2)] 0.16723 0.59883 1 -1 Msboxa [(d5)] 0.88095 0.42857 -1 -1 Msboxa [ 0 0 0 0 ] [ 1 0.666667 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore gsave 0 0 1 setrgbcolor gsave 0.02 setlinewidth 0.34127 0.33333 Mdot [(A)] 0.34127 0.33333 -1 -1 Mshowa grestore 0.004 setlinewidth newpath 0.34127 0.33333 0.31746 0 365.73 arc stroke gsave 0.02 setlinewidth 0.61556 0.49316 Mdot [(it1)] 0.61556 0.49316 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.16723 0.59883 Mdot [(it2)] 0.16723 0.59883 1 -1 Mshowa grestore 1 0 0 setrgbcolor 0.02381 0.63264 moveto 0.97619 0.40816 lineto stroke gsave 0.02 setlinewidth 0.88095 0.42857 Mdot [(d5)] 0.88095 0.42857 -1 -1 Mshowa grestore grestore 0 0 moveto 1 0 lineto 1 0.66667 lineto 0 0.66667 lineto closepath clip newpath % End of Graphics MathPictureEnd\ \>"], "Graphics", ImageSize->{363, 242}, ImageMargins->{{34, Inherited}, {Inherited, Inherited}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False]}, Open]]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Test", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["Descartes-Geometry", FontSlant->"Italic"], " has a number of functions which test the rigorous validity of given \ properties such as alignment, concentricity, parallelism, etc. Here, we \ verify that ", StyleBox["A", FontWeight->"Bold"], " and ", StyleBox["C", FontWeight->"Bold"], " belong to the perpendicular bisector ", StyleBox["d4", FontWeight->"Bold"], " of the segment which joins the contact points ", StyleBox["it", FontWeight->"Bold"], " using the test function ", StyleBox["OfQ", FontWeight->"Bold"], ":" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["OfQ[{a,c},d4]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ True\ \>", "\<\ True\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Images", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Any object may be mirrored (", StyleBox["/", FontWeight->"Bold"], ") in any other object which remains invariant in the operation. The image \ is given by a symmetry when the invariant is a point or a straight line, or \ by an inversion when the invariant is a circle. Let us envisage the three \ cases. " }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["With respect to a point", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The symmetric of ", StyleBox["C", FontWeight->"Bold"], " with respect to ", StyleBox["A", FontWeight->"Bold"], " is ", StyleBox["C'", FontWeight->"Bold"], " defined by" }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ ci1=c/a Draw[Blue,PointSize[.04],{a,\"A\"},{c,\"C\"}, Red,{ci1,\"C'\"}];\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ POINT[-1/3, -2^(1/2)]\ \>", "\<\ 1 POINT[-(-), -Sqrt[2]] 3\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 4.24264 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 1.42857 2.12132 1.42857 [ [(A)] 0.5 2.12132 -1 -1 Msboxa [(C)] 0.97619 4.14163 -1 -1 Msboxa [(C')] 0.02381 0.10102 -1 -1 Msboxa [ 0 0 0 0 ] [ 1 4.24264 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore gsave 0 0 1 setrgbcolor gsave 0.04 setlinewidth 0.5 2.12132 Mdot [(A)] 0.5 2.12132 -1 -1 Mshowa grestore gsave 0.04 setlinewidth 0.97619 4.14163 Mdot [(C)] 0.97619 4.14163 -1 -1 Mshowa grestore 1 0 0 setrgbcolor gsave 0.04 setlinewidth 0.02381 0.10102 Mdot [(C')] 0.02381 0.10102 -1 -1 Mshowa grestore grestore 0 0 moveto 1 0 lineto 1 4.24264 lineto 0 4.24264 lineto closepath clip newpath % End of Graphics MathPictureEnd\ \>"], "Graphics", ImageSize->{84, 351}, ImageMargins->{{34, Inherited}, {Inherited, Inherited}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False]}, Open]]}, Open]], Cell[CellGroupData[{Cell["With respect to a straightline", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The image of the point", StyleBox[" C", FontWeight->"Bold"], " with respect to the straight line ", StyleBox["d5", FontWeight->"Bold"], " is ", StyleBox["C'", FontWeight->"Bold"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ ci2=c/d5 Draw[Blue,PointSize[.02],{c,\"C\"}, d5,PointSize[0],{POINT[1.7,0.3],\"d5\"}, Red,PointSize[.02],{ci2,\"C'\"}, Interval->{{-1,2},{-1,2}}];\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ POINT[-1/57, -2^(1/2)/19]\ \>", "\<\ 1 -Sqrt[2] POINT[-(--), --------] 57 19\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}], Cell[GraphicsData["PostScript", "\<\ %! 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