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Question
If a : b : : c : d, prove that (la + mb) : (lc + mb) :: (la – mb) : (lc – mb)
Solution
∵ a : b :: c : d
∴ `a/b = c/d`
⇒ `"la"/"mb" = "lc"/"md" ...("Multiply by" "l"/"m")`
Applying componendo and dividendo,
`"la + mb"/"la - mb" = "lc + md"/"lc - md"`
⇒ `"la + mb"/"lc + md" = "la - mb"/"lc - md"` ...(By alternendo)
⇒ (la + mb) : (lc + md) :: (la – mb) : (lc – md).
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