解：∵?$BD $?垂直平分?$ AE，$?∴?$AD = DE，$??$BA = BE $?
∵?$ \triangle DEC $?的周長為?$ DE + EC + CD = 6\ \mathrm {cm}$?
∴?$ AD + CD + EC = AC + EC = 6\ \mathrm {cm} $?
?$ \triangle ABC $?的周長為?$ AB + BC + AC = 18\ \mathrm {cm}$?
?$ BC = BE + EC = AB + EC$?
∴?$AB + (AB + EC) + AC = 2\ \mathrm {A}B + (AC + EC) = 18\ \mathrm {cm}$?
∴?$ 2\ \mathrm {A}B + 6 = 18，$?解得?$ AB = 6\ \mathrm {cm} $?
