`△ABC` and `△DBC` are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC. If AD is extended to intersect BC at P, show that (i) `△ABD≅△ACD` (ii) `△ABP≅△ACP` (iii) AP bisects `∠A` as well as `△D`. (iv) AP is the perpendicular bisector of BC.