State and prove Basic Proportionality theorem.

In $∆ABC, AD\perp BC$ and $A{D}^2=BD\times CD$. Prove that $A{B}^2+A{C}^2={\left(BD+CD\right)}^2$.

In $∆ABC, DE\parallel BC,$ If $AD=5 cm, BD=7 cm$ and $AC=18 cm,$ If the length of $AE=k$, then find the value of $k$.

In the given figure if $PQ\parallel RS,$ Prove that $∆POQ~∆SOR.$

Prove that the ratio of the area of two similar triangles is equal to the square of the ratio of their corresponding sides.

In the $△ \mathrm{ABC}$, if $\mathrm{DE} \left|\right| \mathrm{AC}$, then the correct relation is

In the $△ABD$, $C$ is a point on $BD$ such that $BC: CD=1: 2$, and $△ABC$ is an equilateral triangle. Then prove that $A{D}^2=7A{C}^2$.