In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x-3$,$MB = 2x-7$,$AN = x+3$ and $NC = 2x+3$,find the value of $x$.

$A$ $15\, m$ high tower casts a shadow $24\, m$ long at a certain time. At the same time,a telephone pole casts a shadow $16\, m$ long. Find the height of the telephone pole (in $m$).

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB^2 + AC^2 = 122$ and $AD = 6$,find $BC$.

Prove that the sum of the squares of the sides of a rectangle is equal to the sum of the squares of its diagonals.

In quadrilateral $ABCD$,$P$ is the midpoint of $\overline{BC}$. If $\overline{DP}$ produced intersects $\overline{AB}$ produced at $Q$,prove that $AB = 2CD$ given that $ABCD$ is a parallelogram.

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB = 10$,$AC = 24$,and $AD = 13$,find $BC$.