The Cosine Rule for DAPC and DBPC, where Ð ACP=q, gives
AP2
= AC2+PC2-2AC.PC cosq,
PB2
= BC2+PC2-2BC.PC cosq.

Hence
BC2+PC2-PB2
2BC.PC
= AC2+PC2-AP2
2AC.PC
 = cosq.
Hence, multiplying both sides by 2PC/AB, we find that
AP2
AC.AB
 + PC2
AB
 æ
ç
è 		AC-BC
BC.AC
 ö
÷
ø
= PB2
AB.BC
 + AC-BC
AB
 .
As AB+BC=AC, we get the result:
AP2
AB.AC
 + PC2
AC.BC
= 1 + PB2
AB.BC
 .