AB is a line-segment. AX and BY are two equal line-segments drawn on opposite sides of line AB such that AX || BY. If AB and XY intersect each other at P. Prove that:
(i)    ∆APX ≅ ∆BPY
(ii)    AB and XY bisect each other at P.

In figure given below, AD is the median of ∆ABC.
BE ⊥ AD, CF ⊥ AD. Prove that BE = CF.

In figure, OA = OB and OD = OC. Show that:
(i) ∆AOD ≅ ∆BOC and (ii) AD = BC.

Line-segment AB is parallel to another line-segment CD. O is the mid-point of AD (see figure). Show that: (i) ∆AOB ≅ ∆DOC (ii) O is also the mid-point of BC.