[tex] \triangle BCA\sim\triangle DEA\ (AA)\\\text{therefore the ratios of the lengths of their corresponding sides are equal}\\\\\dfrac{BC}{AC}=\dfrac{DE}{AE}\ \ \ |\text{cross multiply}\\\\BC\cdot AE=AC\cdot DE\ \ \ \ |:BC\\\\AE=\dfrac{AC\cdot DE}{BC}\ \ \ \ |:AC\\\\\dfrac{AE}{AC}=\dfrac{DE}{BC}\to\dfrac{DE}{BC}=\dfrac{AE}{AC} [/tex]
