The Cosine Rule for ∆APC and ∆BPC, where ∠ ACP=θ, gives

AP²

= AC²+PC²−2AC.PC cosθ,

PB²

= BC²+PC²−2BC.PC cosθ.

Hence

BC²+PC²−PB²
2BC.PC

= AC²+PC²−AP²
2AC.PC
= cosθ.

Hence, multiplying both sides by 2PC/AB, we find that

AP²
AC.AB
+ PC²
AB

 AC−BC
BC.AC



= PB²
AB.BC
+ AC−BC
AB
.

As AB+BC=AC, we get the result:

AP²
AB.AC
+ PC²
AC.BC

= 1 + PB²
AB.BC
.