In the following figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is parallelogram and $\mathrm{B}\mathrm{C}$ is produced to a point $\mathrm{Q}$ such that $\mathrm{A}\mathrm{D}=\mathrm{C}\mathrm{Q}.$ If $\mathrm{A}\mathrm{Q}$ intersect $\mathrm{D}\mathrm{C}$ at $\mathrm{P},$ show that $\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{B}\mathrm{P}\mathrm{C}\right)=\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{D}\mathrm{P}\mathrm{Q}\right).$

In the following figure, $\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{B}\mathrm{D}\mathrm{E}$ are two equilateral triangles such that $\mathrm{D}$ is the mid-point of $\mathrm{B}\mathrm{C}.$ If $\mathrm{A}\mathrm{E}$ intersects $\mathrm{B}\mathrm{C}$ at $\mathrm{F},$ show that

 
$\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{B}\mathrm{D}\mathrm{E}\right)=\frac{1}{4}\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{A}\mathrm{B}\mathrm{C}\right)$.

In the following figure, $\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{B}\mathrm{D}\mathrm{E}$ are two equilateral triangles such that $\mathrm{D}$ is the mid-point of $\mathrm{B}\mathrm{C}.$ If $\mathrm{A}\mathrm{E}$ intersects $\mathrm{B}\mathrm{C}$ at $\mathrm{F},$ show that

$\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{B}\mathrm{D}\mathrm{E}\right)=\frac{1}{2}\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{B}\mathrm{A}\mathrm{E}\right)$

In the following figure, $\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{B}\mathrm{D}\mathrm{E}$ are two equilateral triangles such that $\mathrm{D}$ is the mid-point of $\mathrm{B}\mathrm{C}.$ If $\mathrm{A}\mathrm{E}$ intersects $\mathrm{B}\mathrm{C}$ at $\mathrm{F},$ show that

$\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{A}\mathrm{B}\mathrm{C}\right)=2\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{B}\mathrm{E}\mathrm{C}\right)$

In the following figure, $\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{B}\mathrm{D}\mathrm{E}$ are two equilateral triangles such that $\mathrm{D}$ is the mid-point of $\mathrm{B}\mathrm{C}.$ If $\mathrm{A}\mathrm{E}$ intersects $\mathrm{B}\mathrm{C}$ at $\mathrm{F},$ show that

$\mathrm{a}\mathrm{r}\left(\mathrm{B}\mathrm{F}\mathrm{E}\right)=\mathrm{a}\mathrm{r}\left(\mathrm{A}\mathrm{F}\mathrm{D}\right)$

In the following figure, $\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{B}\mathrm{D}\mathrm{E}$ are two equilateral triangles such that $\mathrm{D}$ is the mid-point of $\mathrm{B}\mathrm{C}.$ If $\mathrm{A}\mathrm{E}$ intersects $\mathrm{B}\mathrm{C}$ at $\mathrm{F},$ show that

$\mathrm{a}\mathrm{r}\left(\mathrm{B}\mathrm{F}\mathrm{E}\right)=2\mathrm{a}\mathrm{r}\left(\mathrm{FED}\right)$

In the following figure, $\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{B}\mathrm{D}\mathrm{E}$ are two equilateral triangles such that $\mathrm{D}$ is the mid-point of $\mathrm{B}\mathrm{C}.$ If $\mathrm{A}\mathrm{E}$ intersects $\mathrm{B}\mathrm{C}$ at $\mathrm{F},$ show that

 
$\mathrm{a}\mathrm{r}\left(△\mathrm{F}\mathrm{E}\mathrm{D}\right)=\frac{1}{8}\mathrm{a}\mathrm{r}\left(△\mathrm{A}\mathrm{F}\mathrm{C}\right)$

Diagonals $\mathrm{A}\mathrm{C}$ and $\mathrm{B}\mathrm{D}$ of a quadrilateral $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ intersect each other at $\mathrm{P}.$ Show that $\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{A}\mathrm{P}\mathrm{B}\right)\times \mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{C}\mathrm{P}\mathrm{D}\right)=\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{A}\mathrm{P}\mathrm{D}\right)\times \mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{B}\mathrm{P}\mathrm{C}\right)$.

$\mathrm{P}$ and $\mathrm{Q}$ are respectively the mid-points of sides $\mathrm{A}\mathrm{B}$ and $\mathrm{B}\mathrm{C}$ of a triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{R}$ is the mid-point of $\mathrm{A}\mathrm{P},$ show that

$\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{P}\mathrm{R}\mathrm{Q}\right)=\frac{1}{2}\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{A}\mathrm{R}\mathrm{C}\right)$.