In parallel combination of resistances r2 and r3, potential difference will be equal across both resistance.
So, \(i_2r_2 = i_3r_3\)
⇒ \(i_2 = \frac{i_3r_3}{r_2}\) ......(1)
As per Kirchhoff's first law,
\(i_1 = i_2 + i_3\)
\(i_1 = (\frac{r_3}{r_2} + 1)i_3\) (From eq. 1)
⇒ \(\frac{i_3}{i_1}= \frac{r_2}{r_2 + r_3}\)