Solution :
General circle equation :
x^2+y^2+Ax+Bx+C=0 [0]
passes through point A(2,3) means :
2^2+3^2+2A+3B+C=0
2A+3B+C+13=0 [1]
passes through point B(4,5) means :
4^2+5^2+4A+5B+C=0
4A+5B+C+41=0 [2]
passes through point C(2,5) means :
2^2+5^2+2A+5B+C=0
2A+5B+C+29=0 [3]
from [1] and [3] we get :
2A+3B+C+13=0
2A+5B+C+29=0
------------ -
0-2B-16=0
2B=-16
B=-8 [4]
from [2] and [3] :
4A+5B+C+41=0
2A+5B+C+29=0
------------ -
2A-12=0
2A=12
A=6 [5]
substitute [4] and [5] to [1] to be :
2*6+3*(-8)+C+13=0
12-24+C+13=0
C=-1 [6]
from [4], [5], [6] and [0]
we get the answer (circle equation) :
x^2+y^2+6x-8y-1=0