Question

$\frac{x + y}{a + b} = \frac{y + z}{b + c} = \frac{z + x}{c + a} \: \: \frac{x + y + z}{a + b + c}$
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Answer 1

Step-by-step explanation:

Let x/b+c=y/c+a=z/a+b = k

then

x=(b+c)k        ...(1)

y=(c+a)k        ...(2)

z=(a+b)k        ...(3)

to prove

(b-c) x+(c-a) y+(a-b) z=0

L.H.S. = (b-c) x+(c-a) y+(a-b) z

= (b-c) (b+c) k + (c-a) (c+a)k + (a-b) (a+b)k                  by eq (1), (2) & (3)

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

=k[b^2 – c^2 + c^2 – a^2 + a^2 – b^2]

= k [0]

= 0

=R.H.S.