A is symmetric iff A = A. A diagonal matrix is symmetric. E and D are diagonal and symmetric: E = E and D = D. C = C by hypothesis. The product of symmetric matrices is not generally symmetric! (XY) = YX.
(ECE) = ((EC)E) = E(EC) = E(CE) = ECE = ECE.
(ECE) = ECE.
Q.E.D.