(i) In △ABD and △ACD,
AD=AD [ Common side ]
AB=AC [ Given ]
BD=CD [ Given ]
∴ △ABD≅△ACD [ Bu SSS congruence rule ]
∠BAD=∠CAD [ CPCT ] ------ ( 1 )
i.e.∠BAP=∠CAP --------- ( 2 )
(ii) In △ABP and △ACP,
AP=AP [ Common side ]
∠BAP=∠CAP [ From ( 2 ) ]
AB=AC [ Given ]
∴ △ABP≅ACP [ By SAS Congruence rule ]
BP=CP [ By CPCT ] --------- ( 3 )
(iii) ∠BAD=∠CAD [ From ( 1 )]
∴ AP bisects ∠A.
In △BPD and △CPD,
PD=PD [ Common side ]
BD=CD [ Given ]
BP=CP [ From ( 3 ) ]
∴ △BPD≅△CPD [ By SSS Congruence rule ]
∠BDP=∠CDP [ By CPCT ] ------- ( 4 )
(iv) ∠BPD=∠CPD [ From ( 4 ) ]
BP=CP [ From ( 3 ) ]
∠BPD+∠CPD=180o
2∠BPD=180o
∴ ∠BPD=90o
∴ AP is the perpendicular bisector of BC.