(*  given 3 line-lengths, construct a triangle  *)
(*  Calculate triangle apex with intersecting circles  *)
(*  Use Polygon() instead of Triangle()  *)

hyps = 5;
longs = 4;
shorts = 3;

  (*  Discover one angle of triangle by Newton's method.  Assume acute angle  *)
Clear(t);
answ = Replace(t, FindRoot(Cos(t)*longs+Sin(t)*shorts == hyps, {t, 1/Sqrt(2)}));
k = {{0, 0}, {hyps, 0}, {Cos(answ) * longs, Sin(answ) * longs}};
k = Append(k, Part(k, 1));

Graphics({
    Blue, Circle({0, 0}, longs),
    Red, Circle({hyps, 0}, shorts),
    Green, Line(k)
  }, PlotRange -> {-1 - longs, 1 + longs}
)


(*  given 3 line-lengths, construct a triangle  *)
(*  Calculate triangle apex with intersecting circles  *)
(*  Use Polygon() instead of Triangle()  *)

hyps = 5;
longs = 4;
shorts = 3;

  (* Discover angle by dividing into 2 right triangles, then trigonometry  *)
Clear(x);
answ = Replace({x, height}, Flatten(Solve(
  longs / hyps == x / longs &&  (*  cosine x, unknown 'long'  *)
  shorts / hyps == height / longs &&  (*  sine height, unknown 'short'  *)
  shorts / longs == height / x,  (*  tangent longs, known hypotenuse   *)
  {x, height})), 1);
k = {{0, 0}, {hyps, 0}, {Part(answ, 1), Part(answ, 2)}};
k = Append(k, Part(k, 1));

Graphics({
    Blue, Circle({0, 0}, longs),
    Red, Circle({hyps, 0}, shorts),
    Green, Line(k)
  }, PlotRange -> {-1 - longs, 1 + longs}
)