Question

If $\frac{x}{y+z-x}=\frac{y}{z+x-y}=\frac{z}{x+y-z}$ and x + y + z $\ne$0 then show that each ratio is equal to 1.

Answer 1

$$\begin{gathered} \mathrm{Let}: \\ \frac{x}{y+z-x}=\frac{y}{z+x-y}=\frac{z}{x+y-z}=k \\ \mathrm{By}\mathrm{the}\mathrm{theorem}\mathrm{on}\mathrm{equal}\mathrm{ratios},\mathrm{we}\mathrm{get}: \\ k=\frac{x+y+z}{y+z-x+z+x-y+x+y-z} \\ =\frac{x+y+z}{x+y+x} \\ =1 \\ \therefore \frac{x}{y+z-x}=\frac{y}{z+x-y}=\frac{z}{x+y-z}=1 \end{gathered}$$