Question

If (a + b + c + d) (a – b – c + d) = (a + b – c – d) (a – b + c – d), prove that a : b = c : d.

Show, that a, b, c, d are in proportion if: (a + b + c + d) (a – b – c + d) = (a + b – c – d) (a – b + c – d)

Answer 1

Given, `(a + b + c + d)/(a + b - c - d) = (a - b + c - d)/(a - b - c + d)`

Applying componendo and dividendo,

`((a + b + c + d) + (a + b - c - d))/((a + b + c + d) - (a + b - c - d)) = ((a - b + c - d) + (a - b - c + d))/((a - b + c - d) - (a - b - c + d))`

`(2(a + b))/(2(c + d)) = (2(a - b))/(2(c - d))`

`(a + b)/(c + d) = (a - b)/(c -d)`

`(a + b)/(a - b) = (c + d)/(c - d)`

Applying componendo and dividendo,

`(a + b + a - b)/(a + b - a - b) = (c + d + c - d)/(c + d - c + d )`

`(2a)/(2b) = (2c)/(2d)`

`a/b = c/d`

Answer 2

We have `a/b = c/d`

Applying componendo and dividendo

⇒ `(a + b)/(a - b) = (c + d)/(c - d)`

Applying alternendo

⇒ `(a + b)/(c + d) = (a - b)/(c - d)`

Again, applying componendo and dividendo

`(a + b + c + d)/(a + b - c - d) = (a - b + c - d)/(a - b - c + d)`

⇒ (a + b + c + d) (a – b – c + d)

= (a + b – c – d) (a – b + c – d).

Hence proved.