Algebraic ratio

Let's assume $\dfrac a{b+c} = \dfrac b{a+c} = \dfrac c{a+b} = k$

Now

• $\dfrac a{b+c} = k \implies a = kb+kc ~~~......(1)$
• $\dfrac b{a+c} = k \implies b = ka + kc ~~~......(2)$
• $\dfrac c{a+b} = k \implies c = ka + kb ~~~.....(3)$

$(1) + (2) + (3):$

$\begin{aligned} a +b + c & = kb + kc + ka + kc + ka + kb \\ \Rightarrow a + b + c & = k(a+b + c + a+ b+ c) \\ \Rightarrow a + b + c & = k(2a + 2b +2c) \\ \Rightarrow \dfrac{a + b + c}{2(a + b + c)} & = k \\ \Rightarrow k & = \dfrac 12 \\ \implies k & = 0.5 \\ \end{aligned}$

$(1) -(2):$

$\begin{aligned} a - c & = kb+kc - ka - kc\\ \Rightarrow a - b & = k(b + c - a -c) \\ \Rightarrow a - b & = k(b - a) \\ \Rightarrow a - b & = - k(a- b) \\ \implies k & = -1 \\ \end{aligned}$

The answer is true.