\(P=\dfrac{ab}{\left(a-c\right)\left(b-c\right)}+\dfrac{bc}{\left(b-a\right)\left(c-a\right)}+\dfrac{ca}{\left(c-b\right)\left(a-b\right)}\)
\(=\dfrac{ab\left(a-b\right)+bc\left(b-c\right)-ca\left(a-c\right)}{\left(a-b\right)\left(b-c\right)\left(a-c\right)}\)
\(=\dfrac{ab\left(a-b\right)+b^2c-bc^2-a^2c+ac^2}{\left(a-b\right)\left(b-c\right)\left(a-c\right)}\)
\(=\dfrac{ab\left(a-b\right)-c\left(a-b\right)\left(a+b\right)+c^2\left(a-b\right)}{\left(a-b\right)\left(b-c\right)\left(a-c\right)}\)
\(=\dfrac{ab-c\left(a+b\right)+c^2}{\left(b-c\right)\left(a-c\right)}=\dfrac{ab-bc+c^2-ca}{\left(b-c\right)\left(a-c\right)}\)
\(=\dfrac{b\left(a-c\right)-c\left(a-c\right)}{\left(b-c\right)\left(a-c\right)}=\dfrac{\left(b-c\right)\left(a-c\right)}{\left(b-c\right)\left(a-c\right)}=1\)
