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