Prove CE's circle

C1 : x^2 + y^2 + Dx + Ey + F = 0
C2 : x^2 + y^2 + dx + ey + f = 0
let (m,n,), (p,q) be intersection points of C1 and C2
(m,n,), (p,q) on C1 and C2,
so, m^2 + n^2 + Dm + En + F = 0 ------------ (1)
m^2 + n^2 + dm + en + f = 0 ------------ (2)
(1) - (2), (D-d)m + (E-e)n + (F-f) = 0 ------------- (3)
same reason, (D-d)p + (E-e)q + (F-f) = 0 ------------ (4)
consider straight line : (D-d)x + (E-e)y + (F-f) = 0
from (3) and (4), we get (m,n,), (p,q) on the straight line
so, straight line passes intersection points of C1 and C2
straight line is common chord