Exercise 6.6

In the figure, $\mathrm{}\mathrm{D}$ is a point on hypotenuse $\mathrm{}\mathrm{A}\mathrm{C}$ of $∆\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}$, such that $\mathrm{B}\mathrm{D}\perp \mathrm{A}\mathrm{C},\mathrm{D}\mathrm{M}\perp \mathrm{B}\mathrm{C}$ and $\mathrm{D}\mathrm{N}\perp \mathrm{A}\mathrm{B}$. Prove that:
 ${\mathrm{DN}}^2=\mathrm{DM}·\mathrm{AN}$

In the given figure, $\mathrm{}\mathrm{A}\mathrm{D}\mathrm{}$ is a median of a triangle $\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{}\mathrm{A}\mathrm{M}\perp \mathrm{B}\mathrm{C}$. Prove that $\mathrm{A}{\mathrm{B}}^2=\mathrm{A}{\mathrm{D}}^2-\mathrm{B}\mathrm{C}·\mathrm{D}\mathrm{M}+{\left(\frac{\mathrm{B}\mathrm{C}}{2}\right)}^2$

In the figure, $\mathrm{}\mathrm{A}\mathrm{D}\mathrm{}$ is the median of triangle $\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{}\mathrm{A}\mathrm{M}\perp \mathrm{B}\mathrm{C}$. Prove that:
$\mathrm{A}{\mathrm{C}}^2+\mathrm{A}{\mathrm{B}}^2=2\mathrm{A}{\mathrm{D}}^2+\frac{1}{2}\mathrm{B}{\mathrm{C}}^2$

In below figure, two chords $\mathrm{}\mathrm{A}\mathrm{B}$ and $\mathrm{}\mathrm{C}\mathrm{D}$ intersect each other at the point $\mathrm{}\mathrm{P}$. Prove that $△\mathrm{APC}~△\mathrm{DPB}$

In figure two chords $\mathrm{}\mathrm{A}\mathrm{B}$ and $\mathrm{}\mathrm{C}\mathrm{D}$ intersect each other at the point $\mathrm{}\mathrm{P}$. Prove that $\mathrm{A}\mathrm{P}·\mathrm{P}\mathrm{B}=\mathrm{C}\mathrm{P}·\mathrm{D}\mathrm{P}$.

In the figure, two chords $\mathrm{}\mathrm{A}\mathrm{B}\mathrm{}$ and $\mathrm{}\mathrm{C}\mathrm{D}$ of a circle intersect each other at the point $\mathrm{}\mathrm{}\mathrm{P}$ (when produced) outside the circle. Prove that $△PAC~△PDB$.

In the figure, two chords $\mathrm{}\mathrm{A}\mathrm{B}\mathrm{}$ and $\mathrm{}\mathrm{C}\mathrm{D}$ of a circle intersect each other at the point $\mathrm{}\mathrm{}\mathrm{P}$ (when produced) outside the circle. Prove that
$\mathrm{P}\mathrm{A}.\mathrm{P}\mathrm{B}=\mathrm{P}\mathrm{C}.\mathrm{P}\mathrm{D}$

Nazima is fly fishing in a stream. The tip of her fishing rod is $1.8 \mathrm{m}$ above the surface of the water and the fly at the end of the string rests on the water $3.6 \mathrm{m}$ away and $2.4 \mathrm{m}$ from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, how much string does she have out (see the given figure)? If she pulls in the string at the rate of $5 \mathrm{cm}$ per second, what will be the horizontal distance of the fly from her after$12$ seconds?