(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-4, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R32768, L-4, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B32768, L-4, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = leftheader, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = leftfooter, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Overview of Descartes-Geometry by Denis Monasse Mathsoft Overseas :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Availability :[font = text; inactive; preserveAspect; endGroup] Descartes is a registered trade mark of Mathsoft Overseas, Inc. Copyright Ó 1992, by Mathsoft Overseas, Inc. The program Descartesâ-Geometry is available on any platform at 30 % of the Mathematica price. For information and orders, please contact: Mathsoft Overseas, cp 641, 1211 Geneva 3 (Switzerland). Tel: 41 22 / 346 52 60. Fax: 41 22 / 346 59 39 ;[s] 7:0,0;74,1;76,2;131,3;132,4;186,5;198,6;354,-1; 7:1,16,12,Times,0,18,0,0,0;1,0,0,Symbol,0,18,0,0,0;1,16,12,Times,0,18,0,0,0;1,0,0,Symbol,0,18,0,0,0;1,16,12,Times,0,18,0,0,0;1,16,12,Times,2,18,0,0,0;1,16,12,Times,0,18,0,0,0; :[font = subsection; inactive; preserveAspect; startGroup] Introduction :[font = text; inactive; preserveAspect; endGroup] In the seventeenth century, two french mathematicians, Pierre de Fermat and RenŽ 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. Let us start :[font = subsection; inactive; preserveAspect; startGroup] Famous theorems and their animation :[font = text; inactive; preserveAspect] We shall illustrate the functions of Descartes-Geometry on famous theorems such as Pappus, Euler circle and Morley. The geometric figure which illustrates the theorem is built frame by frame and this way the animation of the sequence of frames gives a dynamic feeling of the theorem. :[font = subsubsection; inactive; preserveAspect; startGroup] Pappus :[font = text; inactive; preserveAspect] Let us consider - the points A and B, the straight line d1 passing through A and B, P a point on d1, - a point C and a straight line s passing through C, - the straight lines PC and AC. E is the intersection with s of the straight line passing through B and parallel to PC, F is the intersection with s of the straight line passing through B and parallel to AC, then AE is parallel to PF. We have used the points A, B, C previously defined. P and s are randomly generated so that many different configurations can be contemplated by activating the same cell. ;[s] 46:0,0;30,1;31,2;36,3;37,4;57,5;59,6;76,7;77,8;82,9;83,10;85,11;86,12;98,13;100,14;113,15;114,16;135,17;136,18;153,19;154,20;177,21;179,22;184,23;186,24;188,25;189,26;215,27;216,28;255,29;256,30;274,31;276,32;279,33;280,34;306,35;307,36;346,37;347,38;365,39;367,40;375,41;377,42;393,43;395,44;397,45;572,-1; 46:1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect] p=ParameterPoint[d1,Random[]]; s=c.POINT[]; pc=p.c; ac=a.c; bpc=b.pc; bac=b.ac; e=bpc*s; f=bac*s; range= {{Min[a[[1]],b[[1]],c[[1]],p[[1]],e[[1]],f[[1]]]-1, Max[a[[1]],b[[1]],c[[1]],p[[1]],e[[1]],f[[1]]]+1}, {Min[a[[2]],b[[2]],c[[2]],p[[2]],e[[2]],f[[2]]]-1, Max[a[[2]],b[[2]],c[[2]],p[[2]],e[[2]],f[[2]]]+1}}//N; l[0]={Blue,PointSize[.02],{a,"A"},{b,"B"},d1}; ae=a.e; pf=p.f; l[1]={{c,"C"},s}; l[2]={{p,"P"},pc,ac}; l[3]={Green,{e,"E"},{f,"F"},bpc,bac}; l[4]={Red,ae,pf}; Draw[#, Interval->range, PlotRange->range]&/@ FoldList[Join[#1,#2]&,l[0],Array[l,4]] :[font = text; inactive; preserveAspect] This is a first configuration :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 185; pictureHeight = 293; startGroup; animationSpeed = 104] %! 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%%Creator: Mathematica %%AspectRatio: 1.28142 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.332893 0.332893 0.477744 0.332893 [ [ 0 0 0 0 ] [ 1 1.28142 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 1.28142 lineto 0 1.28142 lineto closepath clip newpath gsave gsave 0 0 1 setrgbcolor gsave 0.02 setlinewidth 0.33289 0.47774 Mdot [(A)] 0.33289 0.47774 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.66579 0.47774 Mdot [(B)] 0.66579 0.47774 -1 -1 Mshowa grestore 0.004 setlinewidth 0 0.47774 moveto 1 0.47774 lineto stroke gsave 0.02 setlinewidth 0.44386 0.94853 Mdot [(C)] 0.44386 0.94853 -1 -1 Mshowa grestore 0.32314 1.28142 moveto 0.78783 0 lineto stroke gsave 0.02 setlinewidth 0.44815 0.47774 Mdot [(P)] 0.44815 0.47774 -1 -1 Mshowa grestore stroke 0.44082 1.28142 moveto 0.4525 0 lineto stroke 0.4525 0 moveto 0 0 rlineto 0.22029 0 moveto 0 0 rlineto 0.52232 1.28142 moveto 0.22029 0 lineto stroke stroke 0 1 0 setrgbcolor gsave 0.02 setlinewidth 0.66711 0.33289 Mdot [(E)] 0.66711 0.33289 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.64561 0.39216 Mdot [(F)] 0.64561 0.39216 -1 -1 Mshowa grestore stroke 0.65846 1.28142 moveto 0.67014 0 lineto stroke 0.67014 0 moveto 0 0 rlineto 0.55318 0 moveto 0 0 rlineto 0.85521 1.28142 moveto 0.55318 0 lineto stroke stroke 1 0 0 setrgbcolor 0 0.62202 moveto 1 0.18861 lineto stroke 0 0.67198 moveto 1 0.23857 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = subsubsection; inactive; preserveAspect; startGroup] Euler circle :[font = text; inactive; preserveAspect] In any triangle, nine points: - the pedals, - the mid-points of each side, - the mid-points of the segments which join each vertex to the orthocenter, are located on a same circle The figure is based on the geometric functions used for the triangle ABC. The pedals are the projections (>) of each vertex onto the opposite sides: ;[s] 6:0,0;182,1;254,2;257,3;291,4;292,5;334,-1; 6:1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect] pr=Table[m[i]>d[i],{i,0,2}]; ht=Table[LINE[m[i],pr[[i+1]]],{i,0,2}]; oc=Orthocenter@@tr; :[font = text; inactive; preserveAspect] The next three points are the middles (MidPoint) of each side: ;[s] 3:0,0;39,1;47,2;64,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect] mp1=Table[MidPoint[m[i],m[Mod[i+1,3]]],{i,0,2}]; :[font = text; inactive; preserveAspect] The last three points are the middles of the segments which join each vertex to the orthocenter (Orthocenter) of the triangle ;[s] 3:0,0;97,1;108,2;126,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect] mp2=Table[MidPoint[m[i],oc],{i,0,2}]; :[font = text; inactive; preserveAspect] The nine point circle is defined as the circle circumscribed (CircumCircle) to the pedals. It is drawn with the triangle and the nine points: ;[s] 3:0,0;61,1;75,2;143,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; startGroup] range= {{Min[a[[1]],b[[1]],c[[1]]]-1, Max[a[[1]],b[[1]],c[[1]]]+1}, {Min[a[[2]],b[[2]],c[[2]]]-1, Max[a[[2]],b[[2]],c[[2]]]+1}}//N; l[0]={Black,{a,"A",{1,-1}},{b,"B"},{c,"C"},tr}; l[1]={Blue,PointSize[.02],{oc,"H"},pr,ht}; l[2]={Green,mp1}; l[3]={Red,mp2}; l[4]={Blue,CircumCircle@@pr}; Draw[#, PlotRange->range]&/@ FoldList[Join[#1,#2]&,l[0],Array[l,4]]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 254; pictureHeight = 289; startGroup; animationSpeed = 80] %! %%Creator: Mathematica %%AspectRatio: 1.13807 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.333333 0.333333 0.333333 0.333333 [ [ 0 0 0 0 ] [ 1 1.13807 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 1.13807 lineto 0 1.13807 lineto closepath clip newpath gsave gsave 0 0 0 setrgbcolor gsave 0.008 setlinewidth 0.33333 0.33333 Mdot [(A)] 0.33333 0.33333 1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.66667 0.33333 Mdot [(B)] 0.66667 0.33333 -1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.44444 0.80474 Mdot [(C)] 0.44444 0.80474 -1 -1 Mshowa grestore 0.004 setlinewidth 0.33333 0.33333 moveto 0.66667 0.33333 lineto 0.44444 0.80474 lineto 0.33333 0.33333 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 255; pictureHeight = 290; animationSpeed = 80] %! %%Creator: Mathematica %%AspectRatio: 1.13807 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.333333 0.333333 0.333333 0.333333 [ [ 0 0 0 0 ] [ 1 1.13807 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 1.13807 lineto 0 1.13807 lineto closepath clip newpath gsave gsave 0 0 0 setrgbcolor gsave 0.008 setlinewidth 0.33333 0.33333 Mdot [(A)] 0.33333 0.33333 1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.66667 0.33333 Mdot [(B)] 0.66667 0.33333 -1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.44444 0.80474 Mdot [(C)] 0.44444 0.80474 -1 -1 Mshowa grestore 0.004 setlinewidth 0.33333 0.33333 moveto 0.66667 0.33333 lineto 0.44444 0.80474 lineto 0.33333 0.33333 lineto stroke 0 0 1 setrgbcolor gsave 0.02 setlinewidth 0.44444 0.38571 Mdot [(H)] 0.44444 0.38571 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.60606 0.4619 Mdot 0.35088 0.40777 Mdot 0.44444 0.33333 Mdot grestore gsave 0.33333 0.33333 moveto 0.60606 0.4619 lineto stroke 0.66667 0.33333 moveto 0.35088 0.40777 lineto stroke 0.44444 0.80474 moveto 0.44444 0.33333 lineto stroke grestore grestore grestore % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 255; pictureHeight = 290; animationSpeed = 80] %! %%Creator: Mathematica %%AspectRatio: 1.13807 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.333333 0.333333 0.333333 0.333333 [ [ 0 0 0 0 ] [ 1 1.13807 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 1.13807 lineto 0 1.13807 lineto closepath clip newpath gsave gsave 0 0 0 setrgbcolor gsave 0.008 setlinewidth 0.33333 0.33333 Mdot [(A)] 0.33333 0.33333 1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.66667 0.33333 Mdot [(B)] 0.66667 0.33333 -1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.44444 0.80474 Mdot [(C)] 0.44444 0.80474 -1 -1 Mshowa grestore 0.004 setlinewidth 0.33333 0.33333 moveto 0.66667 0.33333 lineto 0.44444 0.80474 lineto 0.33333 0.33333 lineto stroke 0 0 1 setrgbcolor gsave 0.02 setlinewidth 0.44444 0.38571 Mdot [(H)] 0.44444 0.38571 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.60606 0.4619 Mdot 0.35088 0.40777 Mdot 0.44444 0.33333 Mdot grestore gsave 0.33333 0.33333 moveto 0.60606 0.4619 lineto stroke 0.66667 0.33333 moveto 0.35088 0.40777 lineto stroke 0.44444 0.80474 moveto 0.44444 0.33333 lineto stroke grestore 0 1 0 setrgbcolor gsave 0.02 setlinewidth 0.5 0.33333 Mdot 0.55556 0.56904 Mdot 0.38889 0.56904 Mdot grestore grestore grestore % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 256; pictureHeight = 291; animationSpeed = 80] %! %%Creator: Mathematica %%AspectRatio: 1.13807 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.333333 0.333333 0.333333 0.333333 [ [ 0 0 0 0 ] [ 1 1.13807 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 1.13807 lineto 0 1.13807 lineto closepath clip newpath gsave gsave 0 0 0 setrgbcolor gsave 0.008 setlinewidth 0.33333 0.33333 Mdot [(A)] 0.33333 0.33333 1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.66667 0.33333 Mdot [(B)] 0.66667 0.33333 -1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.44444 0.80474 Mdot [(C)] 0.44444 0.80474 -1 -1 Mshowa grestore 0.004 setlinewidth 0.33333 0.33333 moveto 0.66667 0.33333 lineto 0.44444 0.80474 lineto 0.33333 0.33333 lineto stroke 0 0 1 setrgbcolor gsave 0.02 setlinewidth 0.44444 0.38571 Mdot [(H)] 0.44444 0.38571 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.60606 0.4619 Mdot 0.35088 0.40777 Mdot 0.44444 0.33333 Mdot grestore gsave 0.33333 0.33333 moveto 0.60606 0.4619 lineto stroke 0.66667 0.33333 moveto 0.35088 0.40777 lineto stroke 0.44444 0.80474 moveto 0.44444 0.33333 lineto stroke grestore 0 1 0 setrgbcolor gsave 0.02 setlinewidth 0.5 0.33333 Mdot 0.55556 0.56904 Mdot 0.38889 0.56904 Mdot grestore 1 0 0 setrgbcolor gsave 0.02 setlinewidth 0.38889 0.35952 Mdot 0.55556 0.35952 Mdot 0.44444 0.59522 Mdot grestore grestore grestore % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 255; pictureHeight = 290; endGroup; endGroup; endGroup; animationSpeed = 80] %! %%Creator: Mathematica %%AspectRatio: 1.13807 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.333333 0.333333 0.333333 0.333333 [ [ 0 0 0 0 ] [ 1 1.13807 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 1.13807 lineto 0 1.13807 lineto closepath clip newpath gsave gsave 0 0 0 setrgbcolor gsave 0.008 setlinewidth 0.33333 0.33333 Mdot [(A)] 0.33333 0.33333 1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.66667 0.33333 Mdot [(B)] 0.66667 0.33333 -1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.44444 0.80474 Mdot [(C)] 0.44444 0.80474 -1 -1 Mshowa grestore 0.004 setlinewidth 0.33333 0.33333 moveto 0.66667 0.33333 lineto 0.44444 0.80474 lineto 0.33333 0.33333 lineto stroke 0 0 1 setrgbcolor gsave 0.02 setlinewidth 0.44444 0.38571 Mdot [(H)] 0.44444 0.38571 -1 -1 Mshowa grestore gsave 0.02 setlinewidth 0.60606 0.4619 Mdot 0.35088 0.40777 Mdot 0.44444 0.33333 Mdot grestore gsave 0.33333 0.33333 moveto 0.60606 0.4619 lineto stroke 0.66667 0.33333 moveto 0.35088 0.40777 lineto stroke 0.44444 0.80474 moveto 0.44444 0.33333 lineto stroke grestore 0 1 0 setrgbcolor gsave 0.02 setlinewidth 0.5 0.33333 Mdot 0.55556 0.56904 Mdot 0.38889 0.56904 Mdot grestore 1 0 0 setrgbcolor gsave 0.02 setlinewidth 0.38889 0.35952 Mdot 0.55556 0.35952 Mdot 0.44444 0.59522 Mdot grestore 0 0 1 setrgbcolor newpath 0.47222 0.46428 0.13386 0 365.73 arc stroke grestore grestore % End of Graphics MathPictureEnd :[font = subsubsection; inactive; preserveAspect; startGroup] Morley :[font = text; inactive; preserveAspect] Morley's theorem is difficult to prove and concerns a problem which cannot be solved with ruler and compass: the trisection of an angle. It states that: The intersections of the adjacent trisectors of a triangle are the vertices of an equilateral triangle. The construction still uses the method of the geometric functions. The program provides the facility of cutting an angle into n equal parts (XSectors). There are 2(n-1) straight lines: n-1 outer (tx) and n-1 inner (ux) x-sectors. Here n is equal to 3, the last argument of the command after ux. Numerical evaluation (N) has been applied to speed up the calculations. As we are not immediately interested in the drawing of the lines, the option LogicalDraw is set to False and we keep the third element [[3]] of the output list which contains the analytic expression of the trisectors in the form of substitution rules ;[s] 19:0,0;157,1;265,2;394,3;396,4;410,5;418,6;465,7;467,8;484,9;486,10;586,11;587,12;713,13;724,14;735,15;740,16;771,17;776,18;887,-1; 19:1,11,8,Times,0,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect] xs=Table[(XSectors[d[i]//N,d[i+1]//N, {tx[i],ux[i],3}, LogicalDraw->False]//N)[[3]], {i,0,2}]//Flatten; :[font = text; inactive; preserveAspect] The intersections qx are those of the first trisector of an angle with the second trisector of the following angle: ;[s] 3:0,0;18,1;20,2;116,-1; 3:1,11,8,Times,0,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect] qx[i_]:=ux[i//N][1//N]*ux[Mod[i+1,3]//N][2//N]/.xs :[font = text; inactive; preserveAspect] Last comes the preparation of the figure. :[font = input; preserveAspect; startGroup] l[0]={Black,{a,"A",{1,-1}},{b,"B"},{c,"C"},tr}; l[1]={Blue,Table[{LINE[m[i],qx[i],m[Mod[i+2,3]]]}, {i,0,2}]}; l[2]={Red,POLYGON[qx[0],qx[1],qx[2]]}; Draw[#, PlotRange->range]&/@ FoldList[Join[#1,#2]&,l[0],Array[l,2]]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 34; pictureWidth = 253; pictureHeight = 288; startGroup; animationSpeed = 37] %! %%Creator: Mathematica %%AspectRatio: 1.13807 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.333333 0.333333 0.333333 0.333333 [ [ 0 0 0 0 ] [ 1 1.13807 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 1.13807 lineto 0 1.13807 lineto closepath clip newpath gsave gsave 0 0 0 setrgbcolor gsave 0.008 setlinewidth 0.33333 0.33333 Mdot [(A)] 0.33333 0.33333 1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.66667 0.33333 Mdot [(B)] 0.66667 0.33333 -1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.44444 0.80474 Mdot [(C)] 0.44444 0.80474 -1 -1 Mshowa grestore 0.004 setlinewidth 0.33333 0.33333 moveto 0.66667 0.33333 lineto 0.44444 0.80474 lineto 0.33333 0.33333 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 36; pictureWidth = 253; pictureHeight = 288; animationSpeed = 37] %! %%Creator: Mathematica %%AspectRatio: 1.13807 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.333333 0.333333 0.333333 0.333333 [ [ 0 0 0 0 ] [ 1 1.13807 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 1.13807 lineto 0 1.13807 lineto closepath clip newpath gsave gsave 0 0 0 setrgbcolor gsave 0.008 setlinewidth 0.33333 0.33333 Mdot [(A)] 0.33333 0.33333 1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.66667 0.33333 Mdot [(B)] 0.66667 0.33333 -1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.44444 0.80474 Mdot [(C)] 0.44444 0.80474 -1 -1 Mshowa grestore 0.004 setlinewidth 0.33333 0.33333 moveto 0.66667 0.33333 lineto 0.44444 0.80474 lineto 0.33333 0.33333 lineto stroke 0 0 1 setrgbcolor gsave gsave 0.33333 0.33333 moveto 0.44193 0.46819 lineto 0.44444 0.80474 lineto stroke grestore gsave 0.66667 0.33333 moveto 0.48418 0.40554 lineto 0.33333 0.33333 lineto stroke grestore gsave 0.44444 0.80474 moveto 0.51731 0.47346 lineto 0.66667 0.33333 lineto stroke grestore grestore grestore grestore % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 254; pictureHeight = 289; endGroup; endGroup; endGroup; endGroup; endGroup; animationSpeed = 37] %! %%Creator: Mathematica %%AspectRatio: 1.13807 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.333333 0.333333 0.333333 0.333333 [ [ 0 0 0 0 ] [ 1 1.13807 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 1.13807 lineto 0 1.13807 lineto closepath clip newpath gsave gsave 0 0 0 setrgbcolor gsave 0.008 setlinewidth 0.33333 0.33333 Mdot [(A)] 0.33333 0.33333 1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.66667 0.33333 Mdot [(B)] 0.66667 0.33333 -1 -1 Mshowa grestore gsave 0.008 setlinewidth 0.44444 0.80474 Mdot [(C)] 0.44444 0.80474 -1 -1 Mshowa grestore 0.004 setlinewidth 0.33333 0.33333 moveto 0.66667 0.33333 lineto 0.44444 0.80474 lineto 0.33333 0.33333 lineto stroke 0 0 1 setrgbcolor gsave gsave 0.33333 0.33333 moveto 0.44193 0.46819 lineto 0.44444 0.80474 lineto stroke grestore gsave 0.66667 0.33333 moveto 0.48418 0.40554 lineto 0.33333 0.33333 lineto stroke grestore gsave 0.44444 0.80474 moveto 0.51731 0.47346 lineto 0.66667 0.33333 lineto stroke grestore grestore 1 0 0 setrgbcolor 0.44193 0.46819 moveto 0.48418 0.40554 lineto 0.51731 0.47346 lineto 0.44193 0.46819 lineto stroke grestore grestore % End of Graphics MathPictureEnd ^*)