Equation of a circle of radius r and centre (h,k):
(x-h)2+(y-k)2=r2, so we can now plug in the given points:
(4-h)2+(2+k)2=r2, (8-h)2+(10+k)2=r2, (1+h)2+(7+k)2=r2.
(4-h)2+(2+k)2=(8-h)2+(10+k)2,
16-8h+h2+4+4k+k2=64-16h+h2+100+20k+k2,
20-8h+4k=164-16h+20k,
8h-16k=144, h-2k=18, (1) h=2k+18
(8-h)2+(10+k)2=(1+h)2+(7+k)2,
64-16h+h2+100+20k+k2=1+2h+h2+49+14k+k2,
164-16h+20k=2h+14k+50,
18h-6k=114, (2) 3h-k=19
3(2k+18)-k=19, 6k+54-k=19, 5k=-35, k=-7, so h=-14+18=4.
Plug (h,k) into (1+h)2+(7+k)2=r2:
52+0=r2, so r=5 and the equation of the circle is (x-4)2+(y+7)2=25.