These are the equations for the altitudes from the points $A$ and $C$ respectively, not ratios. Sorry for the confusion.

For example, to find the altitude from $A$, first note that the slope of $\overline{BC}$ is $- \dfrac{b}{c}$. Therefore, the altitude from $A$ is given by the line through $(a,0)$ with slope $\dfrac{c}{b}$:$$(y-0) = \frac{c}{b} (x - a) \Rightarrow y = \frac{c}{b}x - \frac{ac}{b}.$$A similar calculation gives the altitude from $C$.