Prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangles described on its diagonal.

$D, E, F$ are mid points of sides $BC, CA, AB$ of $△DEF$. Find the ratio of areas of $△DEF$ and $△ABC$.

$△ABC~△DEF$. $BC=3 \mathrm{cm}, EF=4 \mathrm{cm}$ and area of $△ABC=54 {\mathrm{cm}}^2$. Determine the area $△DEF$.

$ABC$ is a triangle and $PQ$ is a straight line meeting $AB$ in $P$ and $AC$ in $Q$. If $AP=1 \mathrm{cm}$ and $BP=3 \mathrm{cm},$ $AQ=1.5 \mathrm{cm}$ and $CQ=4.5 \mathrm{cm}$. Prove that $ar\left(△APQ\right)=\frac{1}{16}\times ar\left(△ABC\right)$.

In $△ACB, \angle C=90°$ and $CD\perp AB$. Prove that $\frac{B{C}^2}{A{C}^2}=\frac{BD}{AD}$.