Pick a pair of equations and solve the system of those equations, and you will get the vertices.
{y=−x−5x=2⟹(x,y)=(2,−7)
{y=−x−5y=−15x+135⟹(x,y)=(−192,92)
{y=−15x+135x=2⟹(x,y)=(2,115)
We find the center of the circle by solving the system
{(x−2)2+(y+7)2=(x+192)2+(y−92)2(x+192)2+(y−92)2=(x−2)2+(y−115)2
which just says the squared distance from the center to each vertex is the same.
Solving gives the center coordinates (−4910,−125).
Radius of circle is just the distance between (−4910,−125) and any one of the vertices, i.e., √(−4910−2)2+(−125−115)2=23√1310.
We use center-radius form of circle equation, and we have the answer: (x+4910)2+(y+125)2=6877100
If you want a simpler answer, multiply both sides by 100 and get (10x+49)2+(10y+24)2=6877.
Exercise: Fill in the missed details in my solution.