Question

If $\frac{a}{x+y}=\frac{b}{y+z}=\frac{c}{z-x},$ then show that b = a + c.

Answer 1

$$\begin{gathered} \mathrm{Let}: \\ \frac{a}{x+y}=\frac{b}{y+z}=\frac{c}{z-x}=k \\ \Rightarrow a=k\left(x+y\right)...\left(\mathrm{i}\right) \\ \Rightarrow b=k\left(y+z\right)...\left(\mathrm{ii}\right) \\ \Rightarrow c=k\left(z-x\right)...\left(\mathrm{iii}\right) \\ \mathrm{We}\mathrm{have}\mathrm{to}\mathrm{show}\mathrm{that}b=a+c. \\ \mathrm{Now},\mathrm{consider}\mathrm{the}\mathrm{RHS}. \\ a+c=k\left(x+y\right)+k\left(z-x\right)[\mathrm{From}(\mathrm{i})\&(\mathrm{iii}\left)\right] \\ =k\left(x+y+z-x\right) \\ =k\left(y+z\right) \\ =b \\ =\mathrm{LHS} \\ \therefore b=a+c \end{gathered}$$