$\dfrac{AB}{AD}=\dfrac 3 5$
$\dfrac{BD}{BC}=\dfrac{6}{10}=\dfrac{3}{5}$
$\to \dfrac{AB}{AD}=\dfrac{BD}{BC}$ (1)
$\dfrac{BD}{BA}=\dfrac{6}{3}=2$
$\dfrac{DC}{DB}=\dfrac{12}{6}=2$
$\to \dfrac{BD}{BA}=\dfrac{DC}{DB}$ (2)
$\dfrac{CD}{CB}=\dfrac{12}{10}=\dfrac{6}{5}$
$\dfrac{DB}{DA}=\dfrac{6}{5}$
$\to \dfrac{CD}{CB}=\dfrac{DB}{DA}$ (3)
(1)(2)(3) $\to \Delta ABD\sim\Delta BDC$