If (pa + qb) : (pc + qd) : : (pa - qb) : (pc - qd), prove that a : b : : c : d.

Given, (pa + qb) : (pc + qd) : : (pa - qb) : (pc - qd).

$\Rightarrow \dfrac{pa + qb}{pc + qd} = \dfrac{pa - qb}{pc - qd} \\[0.5em]$

By alternendo,

$\Rightarrow \dfrac{pa + qb}{pa - qb} = \dfrac{pc + qd}{pc - qd} \\[0.5em]$

By componendo and dividendo,

$\Rightarrow \dfrac{pa + qb + pa - qb}{pa + qb - pa + qb} = \dfrac{pc + qd + pc - qd}{pc + qd - pc + qd} \\[0.5em] \Rightarrow \dfrac{2pa}{2qb} = \dfrac{2pc}{2qd} \\[0.5em]$

On dividing the equation by $\dfrac{2p}{2q}$,

$\Rightarrow \dfrac{a}{b} = \dfrac{c}{d} \\[0.5em] \Rightarrow a : b : : c : d.$

Hence, proved that a, b, c, d are in proportion.