Finding right triangles in random points

n = 12; (* How many points *)
pts = Table[RandomReal[], {n}, {2}];
Show[
 Graphics[{Red, PointSize[0.01], Point[pts]}],
 ImageSize -> 8*72, PlotRange -> {{0, 1}, {0, 1}}, Frame -> True,
 FrameTicks -> False
 ]

\[Theta]min = 89.0; (* Range of angles to be considered "right" (in \
degrees) *)
\[Theta]max = 91.0;
RightAngles = {};
Do[(
  ptA = pts[[i]];
  ptB = pts[[j]];
  ptC = pts[[k]];

  a = Norm[ptB - ptC];
  b = Norm[ptA - ptC];
  c = Norm[ptA - ptB];

  AngleA = 180/\[Pi] ArcCos[(b^2 + c^2 - a^2)/(2 b c)];
  If[\[Theta]min <=
    AngleA <= \[Theta]max, (AppendTo[RightAngles, {ptA, ptB, ptC}];
    Continue[];)];
  AngleB = 180/\[Pi] ArcCos[(a^2 + c^2 - b^2)/(2 a c)];
  If[\[Theta]min <=
    AngleB <= \[Theta]max, (AppendTo[RightAngles, {ptA, ptB, ptC}];
    Continue[];)];
  AngleC = 180/\[Pi] ArcCos[(a^2 + b^2 - c^2)/(2 a b)];
  If[\[Theta]min <= AngleC <= \[Theta]max,
   AppendTo[RightAngles, {ptA, ptB, ptC}]];
  ), {i, 1, n - 2}, {j, i + 1, n - 1}, {k, j + 1, n}]

RightAngles (* Print the list of triples of vertices that form \
"right-ish" triangles *)
RightAngles // Length (* And how many there are *)

(* Show the "right" triangles overlaid over the random points *)
Show[
 Graphics[{Red, PointSize[0.02], Point[pts]}],
 Table[Graphics[{Thick, Hue[i/Length[RightAngles]],
    Line[{RightAngles[[i, 1]], RightAngles[[i, 2]],
      RightAngles[[i, 3]], RightAngles[[i, 1]]}]}], {i, 1,
   Length[RightAngles]}],
 ImageSize -> 8*72, Frame -> True, FrameTicks -> False,
 PlotRange -> {{0, 1}, {0, 1}}
 ]

(* Create the list of unique side lengths in this set of triangles *)


lengths = {};
Do[(
   AppendTo[lengths, Norm[RightAngles[[i, 1]] - RightAngles[[i, 2]]]];
   AppendTo[lengths, Norm[RightAngles[[i, 1]] - RightAngles[[i, 3]]]];
   AppendTo[lengths, Norm[RightAngles[[i, 2]] - RightAngles[[i, 3]]]];
   ), {i, 1, Length[RightAngles]}];
lengths = Union[lengths, lengths];

(* Here is a set of "special" numbers to search for in ratios of \
these lengths, along with their reciprocals *)
nums = {\[Pi], E, EulerGamma, GoldenRatio, Sqrt[2], Sqrt[3], Sqrt[5],
   Sqrt[6], Sqrt[7], Sqrt[8], Sqrt[10]};

(* Make a list of of side length ratios that "match" the above set of \
special numbers *)
f = 0.01; (* What fractional deviation from the "special" numbers is \
acceptable *)
Specials = {};
Do[(
   r = lengths[[i]]/lengths[[j]];
   Do[(
     x = nums[[k]];
     If[Abs[(r - x)/x] <= f,
      AppendTo[
       Specials, {lengths[[i]], lengths[[j]], r, x, Abs[(r - x)/x]}]];
     If[Abs[(r - (1/x))/(1/x)] <= f,
      AppendTo[
       Specials, {lengths[[i]], lengths[[j]], r, 1/x,
        Abs[(r - (1/x))/(1/x)]}]];
     ), {k, 1, Length[nums]}]
   ), {i, 1, Length[lengths]}, {j, 1, Length[lengths]}];

Specials (* Print the list of "special" side length relationships and \
how accurate they are *)
Specials // Length