If the diagonals of a quadrilateral divide each other proportionally, prove that it is a trapezium.

In the adjoining figure, $DB\perp BC, DE\perp AB$ and $AC\perp BC$. Prove that $\frac{BE}{DE}=\frac{AC}{BC}$.

ln the adjoining figure, $AB\perp BC, DE\perp AC$ and $GF\perp BC$. Prove that$△ADE~△GCF$.

In the adjoining figure, $\angle ACB=90°$ and $CD\perp AB$. Prove that $C{D}^2=BD\times AD$.

In the adjoining figure, $\angle A=90°$ and $AD\perp BC$. If $BD=2 \mathrm{cm}$, $CD=8 \mathrm{cm}$ and $AD=k \mathrm{cm}$, find $k$.

In the adjoining figure, two chords $AB$ and $CD$ intersect each other at the point $P$ (when produced) outside the circle. Prove that $△PAC~△PDB$.

In the adjoining figure, two chords $AB$ and $CD$ intersect each other at the point $P$ (when produced) outside the circle. Prove that $PA\times PB=PC\times PD$.

In the adjoining figure, $CM$ and $RN$ are respectively the medians of $△ABC$ and $△PQR$. If $△ABC~△PQR$, prove that: $△AMC~△PNR$.