How to find intersection points of several circles in Graphics without using Solve[ ]?

Question

For example, I randomly plot some circle in Graphics[]:

Graphics[{Table[{Circle[{RandomInteger[5], RandomInteger[5]}, r]}, {r,
     1, 5}]}]

How to find all the intersections in the picture?

Answer 1

circles = 
  Table[Circle[{RandomInteger[5], RandomInteger[5]}, r], {r, 1, 5}];
pts = Table[
   RegionIntersection[circles[[i]], circles[[j]]], {i, 5}, {j, i - 1}];
Graphics[{circles, PointSize[Large], Red, pts}]

Or

fig = Graphics[
   Table[Circle[{RandomInteger[5], RandomInteger[5]}, r], {r, 1, 5}]];
pts = Graphics`Mesh`FindIntersections[fig, 
   Graphics`Mesh`AllPoints -> False];
Show[fig, Graphics[{PointSize[Large], Red, Point[pts]}]]

Answer 2

I solved this problem for something else a while back. I needed it to be fast, so I used compile.

I started out by finding the exact solution symbolically. I used that result to create the helper functions.

Thanks to George Varnavides for calling my attention to your post.

rootSq = Compile[{{distanceSquared, _Real}, {r1, _Real}, {r2, \
_Real}}, -(distanceSquared - r1^2)^2 + 
    2 (distanceSquared + r1^2) r2^2 - r2^4, 
   CompilationTarget :> "C"];

bxby = Compile[{{distanceSquared, _Real}, {r1, _Real}, {r2, _Real}, \
{dx, _Real}, {dy, _Real}},
   {dx, dy} (distanceSquared + r1^2 - r2^2), CompilationTarget :> "C"];
circleIntersectionsXYExact[circle1 : {{x1_, y1_}, r1_}, 
   circle2 : {{x2_, y2_}, r2_}] := 
  Module[{x, y, dx, dy, denom, distanceSquared, distance, bx, by, 
    rootSquared, root, temp}, dx = x2 - x1;
   dy = y2 - y1;
   distanceSquared = (dx^2 + dy^2);
   distance = Sqrt[distanceSquared];
   rootSquared = rootSq[distanceSquared, r1, r2];
   If[rootSquared < 0, Return[{}]];
   root = Sqrt[rootSquared];
   {bx, by} = bxby[distanceSquared, r1, r2, dx, dy];
   {{x1, y1} + {bx - dy root, by + dx root}/(2 distanceSquared), {x1, 
      y1} + {bx + dy root, by - dx root}/(2 distanceSquared)}];

circles = 
  Table[Circle[{RandomInteger[5], RandomInteger[5]}, r], {r, 1, 5}];

intersections = 
 Flatten[circleIntersectionsXYExact @@@ 
   Subsets[circles /. Circle[x_, r_] :> {x, r}, {2}], 1]

Graphics[{circles, Red, PointSize[Large], Point[intersections]}]