If `(a + b + c + d) : (a + b - c - d) = (a - b + c - d):(a-b-c+d)`, then prove that ` a : b = c :d `.

Explanation

`(a + b + c + d) : (a + b - c - d) = (a - b + c - d) : ( a - b - c + d)`
or, `(a + b+ c+d)/(a + b - c -d) = (a - b + c -d)/(a - b - c +d)`
or, ` (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)`
[ by componendo and dividendo process]
or, `(2 (a + b))/(2(c+d)) = (2(a-b))/(2(c-d)) " or , " (a +b)/(c+d) = (a - b)/(c-d)`
or, ` (a+b)/(a-b)= (c+d)/(c-d)` [by alternative process]
or, `(a + b + a - b)/(a + b - a + b) = (c +d + c-d)/(c + d - c +d)` [by componendo and dividendo process]
or, ` (2a)/(2b) = (2c)/(2d) " or, " a/b = c/d`
` :. a : b = c : d `