Question

I am trying to use computer to show some planar geometry plots. I donot know what software can do this, or whether mathematica can produce such plots easily.

For example, I have the following plot to show.
Given any triangle ABC, let AD be the line bisecting angle BAC and intersecting BC at D. Let M be the midpoint of AD. Let the circle whose diameter is AB intersects CM at F.

How to produce these plots and show the relevant labeling of the points in mma? Is it easy to do? Could someone please give an example, or give some recommendation as to what software is best suited for this purposes?

Many thanks.

Answer 1

I thought I’d show how one might approach this in Mathematica. While not the simplest thing to code, it does have flexibility. Also bear in mind that the author is fairly inept when it comes to graphics, so there might be easier and/or better ways to go about it.

offset[pt_, center_, eps_] := center + (1 + eps)*(pt - center);

pointfunc[{pt_List, center_List, ptname_String}, siz_, 
   eps_] := {PointSize[siz], Point[pt], 
   Inset[ptname, offset[pt, center, eps]]};

Manipulate[Module[
  {plot1, plot2, plot3, siz = .02, ab = bb - aa, bc = cc - bb, 
   ac = cc - aa, cen = (aa + bb)/2., x, y, soln, dd, mm, ff, lens, 
   pts, eps = .15},
  plot1 = ListLinePlot[{aa, bb, cc, aa}];
  plot2 = Graphics[Circle[cen, Norm[ab]/2.]];
  soln = NSolve[{Norm[ac]*({x, y} - aa).ab - 
       Norm[ab]*({x, y} - aa).ac == 
      0, ({x, y} - cc).({-1, 1}*Reverse[bc]) == 0}, {x, y}];
  dd = {x, y} /. soln[[1]];
  mm = (dd + aa)/2;
  soln = NSolve[{({x, y} - cen).({x, y} - cen) - ab.ab/4 == 
      0, ({x, y} - cc).({-1, 1}*Reverse[mm - cc]) == 0}, {x, y}];
  ff = {x, y} /. soln;
  lens = Map[Norm[# - cc] &, ff];
  ff = If[OrderedQ[lens], ff[[1]], ff[[2]]];
  pts = {{aa, cen, "A"}, {bb, cen, "B"}, {cc, cen, "C"}, {dd, cen, 
     "D"}, {ff, cen, "F"}, {mm, cen, "M"}, {cen, ff, "O"}};
  pts = Map[pointfunc[#, siz, eps] &, pts];
  plot3 = Graphics[Join[pts, {Line[{aa, dd}], Line[{cc, mm}]}]];
  Show[plot1, plot2, plot3, PlotRange -> {{-.2, 1.1}, {-.2, 1.2}}, 
   AspectRatio -> Full, Axes -> False]],
 {{aa, {0, 0}}, {0, 0}, {1, 1}, Locator},
 {{bb, {.8, .7}}, {0, 0}, {1, 1}, Locator},
 {{cc, {.1, 1}}, {0, 0}, {1, 1}, Locator}, 
 TrackedSymbols :> None]

Here is a screen shot.

Daniel Lichtblau
Wolfram Research