Question

If `"a"/("b + c") = "b"/("c + a") = "c"/("a + b")` , then prove that a(b - c) + b(c-a) + c (a - b) = 0

Answer 1

`"a"/("b + c") = "b"/("c + a") = "c"/("a + b") = "k"`

a = k(b + c) 

b = k(c+a) 

c= k(a+b) 

a (b - c)+ b (c - a)+ c (a - b) = 0 

LHS 

a (b - c)+ b (c - a)+ c (a - b) 

=  k (b + c )(b - c) + k ( c + a)( c - a) + k (a + b )(a - b) 

= k(b² - C²) + k( c² - a²) + k( a² - b²) 

= kb² - kc² + kc² - ka² + ka² - kb²  

= 0 = RHS

LHS = RHS. Hence, proved.