Step 1:  Understand the question and According to the given information make a imaginary diagram.

Step 2: Note down the equal angles in both the triangles (ΔABC ~ ΔFEG)

            NOTE: It is given that ΔABC ~ ΔFEG.

            So,   \angle A = \angle F ,

                   \angle B = \angle E

                   \angle B = \angle G

Step 3: Find the similar triangles hidden in the figure. (ΔACD ~ ΔFGH).

        NOTE: we know that \angle ACD = \angle FGH (Angle bisector)

                    From this We can write \angle DCB = \angle HGE   

                    (Angle bisector)

                     \angle A = \angle F (Proved above)

                    ∴ ΔACD ~ ΔFGH (By AA similarity criterion)

Step 4: Prove the required ratio

            NOTE:  ΔACD ~ ΔFGH

          So, their corresponding sides are in the same ratio.

             \frac{CD}{GH} = \frac{AC}{FG}  

          Condition (i) proved.

Step 5: Find the similar figures hidden in the figure (ΔDCB ~ ΔHGE)

            NOTE: In ΔDCB and ΔHGE,

                         \angle DCB = \angle HGE  (Proved above)

                         \angle B = \angle E (Proved above)

                      ∴ ΔDCB ~ ΔHGE (By AA similarity criterion)

                      Condition (ii) proved