In the figure AC = BC, ∠DCA = ∠ECB and ∠DBC = ∠EAC. Prove that triangles DBC and EAC are congruent and hence DC = EC.

In ΔACE and ΔBCD, we have
              AC = BC ...... (given)
           󐺬 = ∠DBC ...... (given)
            ∠DCA = ∠ECB   ...... (given)
∠DCA + ∠DCE = ∠ECB + ∠DCE  ........(Adding ∠DCE  on on both sides)
        ⇒ ∠ACE = ∠BCD
            ΔACE ≅ ΔBCD ...... (ASA criterion)
               DC = EC ...... (c.p.c.t.)