Are the inscribed circle equations calculated by these two different lines of code equal?

Question

ClearAll["`*"]
p1 = {3, 5};
p2 = {6, 7};
p3 = {8, 9};
Cir = Insphere[{p1, p2, p3}]
eq1 = SubtractSides[
  Simplify[RegionMember[Cir][{x, y}], {x, y} \[Element] Reals]]

The generated equation is:

-(2/(31 + 2 Sqrt[26] + 2 Sqrt[82] + Sqrt[
    533])) + (-((2 (123 Sqrt[2] + 6 Sqrt[41] + 4 Sqrt[1066]))/(
     41 Sqrt[2] + 4 Sqrt[41] + Sqrt[1066])) + 
    x)^2 + (-((287 Sqrt[2] + 20 Sqrt[41] + 9 Sqrt[1066])/(
     41 Sqrt[2] + 4 Sqrt[41] + Sqrt[1066])) + y)^2 == 0

The equation generated using this line of code is:

Solve[ForAll[{x, y}, 
   Apply[Subtract, eq1] == (x - a)^2 + (y - b)^2 - r^2], {a, b, r}, 
  Assumptions -> r > 0];
(x - a)^2 + (y - b)^2 == r^2 /. %

the result is:

{((1304571024162 + 579809344072 Sqrt[26] - 289904672036 Sqrt[82] - 
      144952336018 Sqrt[533])/289904672036 + 
     x)^2 + ((-5653141104702 - 724761680090 Sqrt[26] + 
      434857008054 Sqrt[82] + 144952336018 Sqrt[533])/289904672036 + 
     y)^2 == (1/
  1500798551219655721166)(1576588878056248335084883 + 
    309164501551249078560196 Sqrt[26] - 
    174092631941480063655256 Sqrt[82] - 
    68286334080494335313053 Sqrt[533])}

Are these two equations generated equal? Why? How to verify?

Answer 1

Taking your two equations and equating them, then simplifying:

FullSimplify[-(2/(31 + 2 Sqrt[26] + 2 Sqrt[82] + 
    Sqrt[533])) + (-((2 (123 Sqrt[2] + 6 Sqrt[41] + 
         4 Sqrt[1066]))/(41 Sqrt[2] + 4 Sqrt[41] + Sqrt[1066])) + 
  x)^2 + (-((287 Sqrt[2] + 20 Sqrt[41] + 
       9 Sqrt[1066])/(41 Sqrt[2] + 4 Sqrt[41] + Sqrt[1066])) + 
  y)^2 == ((1304571024162 + 579809344072 Sqrt[26] - 
     289904672036 Sqrt[82] - 144952336018 Sqrt[533])/
   289904672036 + 
  x)^2 + ((-5653141104702 - 724761680090 Sqrt[26] + 
     434857008054 Sqrt[82] + 144952336018 Sqrt[533])/
   289904672036 + 
  y)^2 - ((1/1500798551219655721166) (1576588878056248335084883 + 
   309164501551249078560196 Sqrt[26] - 
   174092631941480063655256 Sqrt[82] - 
   68286334080494335313053 Sqrt[533]))]

yields True. Note that I had to remove the brackets around the second equation.