(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 267572, 5986]*) (*NotebookOutlinePosition[ 268462, 6017]*) (* CellTagsIndexPosition[ 268372, 6011]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Teaching Geometry to my 12-year old Grandson using ", StyleBox["Geometrica", FontSlant->"Italic"], " " }], "Title"], Cell[TextData[{ StyleBox["Bruno Autin", FontFamily->"Helvetica", FontSize->24, FontWeight->"Bold", FontColor->RGBColor[0.6, 0.4, 0.2]], StyleBox["\nbautin1@club-internet.fr", FontFamily->"Helvetica", FontSize->16, FontColor->RGBColor[0.6, 0.4, 0.2]] }], "Text"], Cell[TextData[{ "This is holiday time and my grandson, Jos, is curious about ", StyleBox["Geometrica", FontSlant->"Italic"], ". I explain a few things, and we agree to survey special lines called ", "perpendicular bisector", StyleBox["s", FontSlant->"Italic"], " or, more usually, ", "bisectors in a triangle", ". First, we learn how to call ", StyleBox["Geometrica", FontSlant->"Italic"], ", then to draw a triangle and determine step by step the bisectors of the \ triangle. We finish with the circumcircle of a triangle. The naive questions \ asked by my grandson shed light on the basic concepts of both ", StyleBox["Mathematica", FontSlant->"Italic"], " and ", StyleBox["Geometrica", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ "Calling ", StyleBox["Geometrica", FontSlant->"Italic"] }], "Section"], Cell[BoxData[ \(<< Geometrica`Geometrica05`\)], "Input", CellLabel->"In[1]:="], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " I trust you, but this is a strange way of using a program.\n", StyleBox["Bruno:", FontWeight->"Bold"], " In ", StyleBox["Mathematica", FontSlant->"Italic"], ", there is a syntax as in any language: English, French, Spanish, Chinese, \ ... The sign \"<<\" means \"Call\". What you call is the program \"", StyleBox["Geometrica", FontSlant->"Italic"], "05\" located in the folder ", StyleBox["Geometrica", FontSlant->"Italic"], ". " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Manipulating a Triangle", "Section"], Cell[TextData[{ StyleBox["Bruno:", FontWeight->"Bold"], " We want to draw a triangle that has no special property. The function to \ do that is ", StyleBox["Triangle[] ", FontFamily->"Courier New"], StyleBox["and we give the name ", FontFamily->"Times New Roman"], StyleBox["t", "MR", FontSlant->"Italic"], StyleBox[" to that triangle.", FontFamily->"Times New Roman"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t = Triangle[]\)], "Input", CellLabel->"In[2]:="], Cell[BoxData[ \(Segment[CPoint[\(-\(\@3\/2\)\), \(-\(1\/2\)\)], CPoint[\@3\/2, \(-\(1\/2\)\)], CPoint[1\/2, \@3\/2], CPoint[\(-\(\@3\/2\)\), \(-\(1\/2\)\)]]\)], "Output", CellLabel->"Out[2]="] }, Open ]], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " I understand ", StyleBox["Triangle", "MR"], " but why the brackets after it?\n", StyleBox["Bruno:", FontWeight->"Bold"], " In ", StyleBox["Mathematica", FontSlant->"Italic"], ", any statement is made of an operator that acts on variables enclosed in \ a pair of brackets. Here, the operator is ", StyleBox["Triangle", FontFamily->"Courier New"], ". The variables of ", StyleBox["Triangle", FontFamily->"Courier New"], " are the three vertices. In our case, they are hidden, we call them \ default variables. You can see what they actually are in the result returned \ by ", StyleBox["Geometrica", FontSlant->"Italic"], ". They are the three points of general form ", StyleBox["CPoint[x,y], ", FontFamily->"Courier New"], "where ", StyleBox["x", "MR"], " and ", StyleBox["y", "MR"], " are the coordinates of the point.\n", StyleBox["Jos:", FontWeight->"Bold"], " What are the coordinates of a point?\n", StyleBox["Bruno:", FontWeight->"Bold"], " To know where a point is in the plane, one defines a reference system \ made of two perpendicular axes ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " that intersect at the origin", StyleBox[" ", FontSlant->"Italic"], "0. The coordinates of a point are the ", "abscissa", " and the ", "ordinate", ". The abscissa is the distance of the point to the ", StyleBox["y", FontSlant->"Italic"], "-axis and the ordinate the distance to the ", StyleBox["x", FontSlant->"Italic"], "-axis.\n", StyleBox["Jos:", FontWeight->"Bold"], " I do not see the point of all that complication. At school, we define a \ triangle by three points. I put the points somewhere on a page of a notebook \ and I join them by lines. Nothing more.\n", StyleBox["Bruno:", FontWeight->"Bold"], " OK, this is the way Greeks dealt with geometry. When one works with a \ computer, we have to define a mechanism that the computer can understand. \ That mechanism is the ", "analytic", " geometry invented by Pierre de Fermat and Ren\[EAcute] Descartes in the \ seventeenth century. But I go too far. You will learn that in more detail \ later on. Now, I shall type a semi-colon at the end of each statement, the \ result will not appear and we shall draw all we need. We continue?\n", StyleBox["Jos:", FontWeight->"Bold"], " No, there is still something bizarre. You defined a triangle and I see ", StyleBox["Segment", FontFamily->"Courier New"], " in the result.\n", StyleBox["Bruno:", FontWeight->"Bold"], " If you give me two points and ask me to join them by a piece of straight \ line, I draw a segment. True?\n", StyleBox["Jos:", FontWeight->"Bold"], " Yes.\n", StyleBox["Bruno:", FontWeight->"Bold"], " If you give me three, four, five points, I shall draw a triangle, a \ quadrangle, a pentagon. And I can continue with as many points as you like. \ The operation is always the same. It consists of joining points with segments \ of straight lines. The figures so obtained belong to the family of the \ polygonal lines. I had to choose a name for the family, it could have been ", StyleBox["PolygonalLine", FontFamily->"Courier New"], " but it was long and not well-adapted for two points. I preferred to take \ the most elementary one, the segment.\n", StyleBox["Jos:", FontWeight->"Bold"], " Fine. Go ahead now and draw me figures.\n", StyleBox["Bruno:", FontWeight->"Bold"], " The simplest thing to do is to draw the triangle ", StyleBox["t", "MR"], " with the function ", StyleBox["Draw", FontFamily->"Courier New"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Draw[t];\)\)], "Input", CellLabel->"In[3]:="], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .78868 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % PlotRegion .05 .95 .05 .95 MPlotRegion % Scaling calculations 0.5 0.549857 0.293707 0.549857 [ [ 0 0 0 0 ] [ 1 .78868 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .5 Mabswid [ ] 0 setdash .02381 .01878 m .97619 .01878 L .77493 .7699 L .02381 .01878 L s 0 0 m 1 0 L 1 .78868 L 0 .78868 L closepath clip newpath % End 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A triangle, I am used to calling it ABC.\n", StyleBox["Bruno:", FontWeight->"Bold"], " For that, you have to add the function ", StyleBox["Legend ", FontFamily->"Courier New"], "to the arguments of ", StyleBox["Draw", FontFamily->"Courier New"], ". 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Open ]], Cell[CellGroupData[{ Cell["Bisectors", "Section"], Cell[TextData[{ StyleBox["Bruno:", FontWeight->"Bold"], " You remember what bisectors are?\n", StyleBox["Jos:", FontWeight->"Bold"], " There are bisectors for segments and for ... angles.\n", StyleBox["Bruno:", FontWeight->"Bold"], " Dealing with angles is quite possible with ", StyleBox["Geometrica", FontSlant->"Italic"], ", but the definition of angles is somewhat involved and that would take us \ too far. Let us limit ourselves to the bisector of a segment that is \ perpendicular to the midpoint of a segment. The segments that we use are the \ sides of the triangle.\n", StyleBox["Jos:", FontWeight->"Bold"], " I will guess how to find them." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Sides[t]\)], "Input", CellLabel->"In[103]:="], Cell[BoxData[ \({Segment[CPoint[\(-\(\@3\/2\)\), \(-\(1\/2\)\)], CPoint[\@3\/2, \(-\(1\/2\)\)]], Segment[CPoint[\@3\/2, \(-\(1\/2\)\)], CPoint[1\/2, \@3\/2]], Segment[CPoint[1\/2, \@3\/2], CPoint[\(-\(\@3\/2\)\), \(-\(1\/2\)\)]]}\)], "Output", CellLabel->"Out[103]="] }, Open ]], Cell[TextData[{ StyleBox["Bruno:", FontWeight->"Bold"], " Bravo. How many sides did you find?\n", StyleBox["Jos:", FontWeight->"Bold"], " Three.\n", StyleBox["Bruno:", FontWeight->"Bold"], " For a start, I would like you to work with one side only, say the first \ one.\n", StyleBox["Jos:", FontWeight->"Bold"], " How can I select it among the three?\n", StyleBox["Bruno:", FontWeight->"Bold"], " The result you got is a list. The three sides are inside a pair of curly \ brackets. Remember these basic notations: a pair of square brackets for \ variables of a function and a pair of curly brackets for a list of objects. A \ neat thing to do is to define the sides by a list of three objects also. The \ first, second and third object of the left-hand list will be the name of the \ first, second and third side of the right-hand list. Let me do it for you." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({s1, s2, s3} = Sides[t]\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \({Segment[CPoint[\(-\(\@3\/2\)\), \(-\(1\/2\)\)], CPoint[\@3\/2, \(-\(1\/2\)\)]], Segment[CPoint[\@3\/2, \(-\(1\/2\)\)], CPoint[1\/2, \@3\/2]], Segment[CPoint[1\/2, \@3\/2], CPoint[\(-\(\@3\/2\)\), \(-\(1\/2\)\)]]}\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " I would like to see what they are.\n", StyleBox["Bruno:", FontWeight->"Bold"], " Draw them in different colors.\n", StyleBox["Jos:", FontWeight->"Bold"], " I will try." }], "Text"], Cell[BoxData[ \(\(Draw[Red, s1, Green, s2, Blue, s3];\)\)], "Input", CellLabel->"In[86]:="], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .78868 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font 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Now, we shall learn how to place a point on a \ segment. There is a general function called ", StyleBox["Pointer", "MR"], " to attach a point to an object. Here the object is the segment ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["s", "MR", FontSlant->"Plain"], "1"], TraditionalForm]]], ". The position of the point on a segment ", StyleBox["AB", "MR"], " is defined by a variable called a ", "parameter", ". If the point is in ", StyleBox["A", "MR"], ", the parameter is 0. If the point is in ", StyleBox["B", "MR"], ", the parameter is 1. Can you guess the value of the parameter for the \ midpoint?\n", StyleBox["Jos:", FontWeight->"Bold"], " 0.5.\n", StyleBox["Bruno:", FontWeight->"Bold"], " Right or ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], ". If you use 0.5, all the results will be given with decimal points. If \ you take ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], ", the results will be exact. As our problem is simple, it is better to \ take ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(m = Pointer[s1, 1/2]\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(CPoint[0, \(-\(1\/2\)\)]\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell["Jos: Let me check.", "Text"], Cell[BoxData[ \(\(Draw[m, Red, s1];\)\)], "Input", CellLabel->"In[90]:="], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .57735 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % PlotRegion .05 .95 .05 .95 MPlotRegion % Scaling calculations 0.5 0.549857 0.563604 0.549857 [ [ 0 0 0 0 ] [ 1 .57735 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g 5 Mabswid .5 .28868 Mdot 1 0 0 r .5 Mabswid [ ] 0 setdash .02381 .28868 m .97619 .28868 L s 0 0 m 1 0 L 1 .57735 L 0 .57735 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Text", CellLabel->"From In[90]:=", ImageSize->{87.375, 50.4375}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg"], ImageRangeCache->{{{4.3125, 82}, {47, 2.5}} -> {-1.0183, -1.08252, \ 0.0235957, 0.0235957}}] }, Open ]], Cell[CellGroupData[{ Cell["First Construct", "Subsection"], Cell[TextData[{ StyleBox["Bruno:", FontWeight->"Bold"], " Now, we have to find the perpendicular to the segment ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["s", "MR", FontSlant->"Plain"], "1"], TraditionalForm]]], " at ", StyleBox["m", FontSlant->"Italic"], ".\n", StyleBox["Jos:", FontWeight->"Bold"], " I know, one takes a compass, places the center at one end of the segment \ and draws an arc of circle around the midpoint. One does the same thing at \ the other end, and we join the intersections of the two arcs.\n", StyleBox["Bruno:", FontWeight->"Bold"], " I will give you the tool. It is the function ", StyleBox["ECircle", FontFamily->"Courier New"], ", its variables are the center of the circle and the radius. You will ask \ me why ", StyleBox["ECircle", FontFamily->"Courier New"], " and not simply ", StyleBox["Circle", FontFamily->"Courier New"], ". This is because ", StyleBox["Circle", FontFamily->"Courier New"], " is already a ", StyleBox["Mathematica", FontSlant->"Italic"], " function that denotes a graphical object, whereas ", StyleBox["ECircle", FontFamily->"Courier New"], " is a geometric object that you can draw, transform, intersect, etc. The \ ", "E", " reminds us that it is a Euclidean definition, from the name of the Greek \ geometer Euclid.\n", StyleBox["Jos:", FontWeight->"Bold"], " I think I can do it. Let me try." }], "Text"], Cell[BoxData[ \({c1, c2} = {ECircle[]}\)], "Input"], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " But I do not have the ends of ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["s", "MR", FontSlant->"Plain"], "1"], TraditionalForm]]], "...\n", StyleBox["Bruno:", FontWeight->"Bold"], " You will learn a great feature of ", StyleBox["Mathematica", FontSlant->"Italic"], ": changing the head of an object. It is very natural to change the \ variables of a function. The name ", "variable", " indicates that they may take any value. It is more surprising to keep the \ variables fixed and change the name of the function. To get the ends of ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["s", "MR", FontSlant->"Plain"], "1"], TraditionalForm]]], ", it would be ideal to replace the head ", StyleBox["Segment", FontFamily->"Courier New"], " by the head ", StyleBox["List", FontFamily->"Courier New"], ". This is actually feasible using the notation @@. Let me show you." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({a, b} = List @@ s1\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \({CPoint[\(-\(\@3\/2\)\), \(-\(1\/2\)\)], CPoint[\@3\/2, \(-\(1\/2\)\)]}\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " It works, but I do not see the word ", StyleBox["List", FontFamily->"Courier New"], ".\n", StyleBox["Bruno:", FontWeight->"Bold"], " It is a matter of notation. The result in the form of curly brackets is \ interpreted by ", StyleBox["Mathematica", FontSlant->"Italic"], " as ", StyleBox["List[a, b]", FontFamily->"Courier New"], ".\n", StyleBox["Jos:", FontWeight->"Bold"], " I believe you. Let me continue my construction now." }], "Text"], Cell[BoxData[ \({c1, c2} = {ECircle[\(a\)\(,\)]}\)], "Input"], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " Ah, but what do I take for the radius?\n", StyleBox["Bruno:", FontWeight->"Bold"], " Evaluate the distance from ", StyleBox["a", "MR"], " to ", StyleBox["b", "MR"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(d = Distance[a, b]\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \(\@3\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " I should be able to do it now." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\({c1, c2} = {ECircle[a, 3 d/4], ECircle[b, 3 d/4]};\)\), "\n", \(\(Draw[s1, Blue, c1, c2];\)\)}], "Input", CellLabel->"In[6]:="], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .6 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def 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It would have been useful if you had drawn the bisector as the line \ perpendicular to ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["s", "MR", FontSlant->"Plain"], "1"], TraditionalForm]]], " in ", StyleBox["m", "MR"], ". Still with ", StyleBox["ELine", FontFamily->"Courier New"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bi1 = ELine[s1, m]\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(ELine[ Segment[CPoint[\(-\(\@3\/2\)\), \(-\(1\/2\)\)], CPoint[\@3\/2, \(-\(1\/2\)\)]], CPoint[0, \(-\(1\/2\)\)]]\)], "Output",\ CellLabel->"Out[11]="] }, Open ]], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " I found a bug in your program!\n", StyleBox["Bruno:", FontWeight->"Bold"], " You used ", StyleBox["ELine", FontFamily->"Courier New"], " before with two points. You reused it with a segment and a point. This \ form is not recognized by ", StyleBox["Geometrica", FontSlant->"Italic"], ". Let us look at what ", StyleBox["Geometrica", FontSlant->"Italic"], " offers to you." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?ELine\)\)], "Input", CellLabel->"In[102]:="], Cell[BoxData["\<\"\[FilledSmallSquare]\!\(\* StyleBox[\\\" \\\",\\n\\\"MR\\\ \"]\)\!\(\* StyleBox[\\\"ELine\\\",\\n\\\"MR\\\"]\)\!\(\* \ StyleBox[\\\"[\\\",\\n\\\"MR\\\"]\)\!\(\* StyleBox[\\\"a\\\",\\n\\\"TI\\\"]\ \)\!\(\* StyleBox[\\\",\\\",\\n\\\"MR\\\"]\)\!\(\* StyleBox[\\\"b\\\",\\n\ \\\"TI\\\"]\)\!\(\* StyleBox[\\\",\\\",\\n\\\"MR\\\"]\)\!\(\* StyleBox[\\\ \"option\\\",\\n\\\"TI\\\"]\)\!\(\* StyleBox[\\\"]\\\",\\n\\\"MR\\\"]\)line \ passing through points \!\(\* StyleBox[\\\"a\\\",\\n\\\"TI\\\"]\)\!\(\* \ StyleBox[\\\" \\\",\\n\\\"MR\\\"]\)and \!\(\* \ StyleBox[\\\"b\\\",\\n\\\"TI\\\"]\).The line is oriented by the vector \!\(\* \ StyleBox[\\\"ab\\\",\\n\\\"TI\\\"]\).\[FilledSmallSquare]\!\(\* StyleBox[\ \\\" \\\",\\n\\\"MR\\\"]\)\!\(\* \ StyleBox[\\\"ELine\\\",\\n\\\"MR\\\"]\)\!\(\* \ 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Better use the function ", StyleBox["IdenticalQ", FontFamily->"Courier New"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(IdenticalQ[bi, bi1]\)], "Input", CellLabel->"In[112]:="], Cell[BoxData[ \(True\)], "Output", CellLabel->"Out[112]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Quick Construct", "Subsection"], Cell[TextData[{ StyleBox["Bruno:", FontWeight->"Bold"], " A program is written not only to emulate what you learn at school but \ also to work quickly. Therefore, you can get the bisector of a segment \ without performing a detailed construction.\n", StyleBox["Jos:", FontWeight->"Bold"], " With ", StyleBox["Bisector", FontFamily->"Courier New"], " and a segment?\n", StyleBox["Bruno:", FontWeight->"Bold"], " Yes." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Bisector[s1]\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(CLine[\(-\@3\), 0, 0]\)], "Output", CellLabel->"Out[12]="] }, Open ]], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " And for a triangle?\n", StyleBox["Bruno:", FontWeight->"Bold"], " And for any polygonal line, remember that the family of the polygonal \ lines was called ", StyleBox["Segment", FontFamily->"Courier New"], " in ", StyleBox["Geometrica", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Bisector[t]\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \({CLine[\(-\@3\), 0, 0], CLine[1\/2\ \((\(-1\) + \@3)\), 1\/2\ \((\(-1\) - \@3)\), \(-\(1\/8\)\)\ \((\(-1\) - \@3)\)\ \ \((\(-1\) + \@3)\) + 1\/8\ \((1 - \@3)\)\ \((1 + \@3)\)], CLine[1\/2\ \((1 + \@3)\), 1\/2\ \((1 + \@3)\), 1\/8\ \((\(-1\) - \@3)\)\ \((1 - \@3)\) - 1\/8\ \((\(-1\) + \@3)\)\ \((1 + \@3)\)]}\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " Cool. Let me draw everything. 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StyleBox["Bruno:", FontWeight->"Bold"], " Yes. ", StyleBox["Geometrica", FontSlant->"Italic"], " can tell you that the conjecture is true. You intersect two bisectors ... \ \n", StyleBox["Jos:", FontWeight->"Bold"], " OK, I know." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\({b1, b2, b3} = bt;\)\), "\[IndentingNewLine]", \(x = Intersections[b1, b2]\)}], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(CPoint[ 0, \(\@3\ \((\(-\(1\/8\)\)\ \((\(-1\) - \@3)\)\ \((\(-1\) + \@3)\) + \ 1\/8\ \((1 - \@3)\)\ \((1 + \@3)\))\)\)\/\(3\/2 + \@3\/2\)]\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[TextData[{ StyleBox["Bruno:", FontWeight->"Bold"], " The result is not pretty, try to simplify it using ", StyleBox["Simplify", FontFamily->"Courier New"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[x]\)], "Input", CellLabel->"In[24]:="], Cell[BoxData[ \(CPoint[0, 0]\)], "Output", CellLabel->"Out[24]="] }, Open ]], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " Bingo!\n", StyleBox["Bruno:", FontWeight->"Bold"], " And now you can check that ", StyleBox["x", "MR"], " belongs to the third bisector using ", StyleBox["OfQ", FontFamily->"Courier New"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(OfQ[x, b3]\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \(True\)], "Output", CellLabel->"Out[9]="] }, Open ]], Cell[TextData[{ StyleBox["Jos:", FontWeight->"Bold"], " Our teacher asks us for more than true or false. He wants a proof.\n", StyleBox["Bruno:", FontWeight->"Bold"], " ", StyleBox["Geometrica", FontSlant->"Italic"], " has an internal procedure of validation based on a solution of equations. \ It does not give you the sequence of statements that makes a proof, but isn't \ it better to leave you the privilege of reasoning?\n", StyleBox["Jos:", FontWeight->"Bold"], " You mean that, even with ", StyleBox["Geometrica", FontSlant->"Italic"], ", I shall still have to do my homework?\n", StyleBox["Bruno:", FontWeight->"Bold"], " Yes. Let me show you one last thing. This point, common to the bisectors, \ is the center of the circle circumscribed to the triangle.\n", StyleBox["Jos:", FontWeight->"Bold"], " I will do that and we will stop because I am a bit tired with Euclid, \ parameters, coordinates and all that stuff. 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