In $\Delta ABC$,$\overline{AD}$ is a median and $AB^2 + AC^2 = 148$. If $AD = 7$,then $BC = \ldots$

Prove that the midpoints of the sides of any convex quadrilateral are the vertices of a parallelogram.

In $\Delta ABC$,$m \angle A = 90^{\circ}$ and $\overline{AD}$ is an altitude to the hypotenuse $BC$. Then,$AD^{2} = \ldots$

In $\Delta ABC$,$m\angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. Prove the following:
$(i)$ The correspondence $AMB \leftrightarrow ABC$ is a similarity.
(ii) The correspondence $BMC \leftrightarrow ABC$ is a similarity.
(iii) The correspondence $AMB \leftrightarrow BMC$ is a similarity.

In $\Delta ABC$ and $\Delta XYZ$,$\frac{AB}{XY} = \frac{BC}{YZ}$ and $\angle B \cong \angle Y$. If $AB = 6$,$AC = 15$,and $XY = 15$,find $XZ$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = 3x - 1$,$MB = 2x + 1$,$AN = 3x + 1$ and $NC = 5x + 1$,find the value of $x$.