Question

the question

Answer 1

the first two ratios, after cross-multiplying:
\[ \frac{a}{b+c}= \frac{b}{a+c} \\ a(a+c)= b(b+c) \]
distribute on both sides:
\[ a^2 +ac= b^2 +bc \\
a^2 -b^2 = bc-ac \]
factor the left side.
\[ (a+b)(a-b)= (b-a)c \\
(a+b)(a-b) = - (a-b) c \]
for \(a \ne b) we can divide both sides:
\[ a+b = -c \\
a+b+c =0 \]

Answer 2

how do we use that result?
we see
\[ c = -(a+b) \\
\frac{c}{a+b} = - \frac{(a+b)}{(a+b)} \\
\frac{c}{a+b} = -1\]

Answer 3

thnks