In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at point $D$,then

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $D$ is a point on $\overline{BC}$. Prove that $AD^{2} + BC^{2} = AC^{2} + BD^{2}$.

In $\Delta XYZ$,$m \angle Y = 90^\circ$ and $\overline{YM}$ is an altitude to the hypotenuse $\overline{XZ}$. If $XM = 5$ and $ZM = 4$,find $YZ$.

In $\Delta ABC$,$B-D-C$. The bisectors of $\angle ADB$ and $\angle ADC$ intersect $\overline{AB}$ and $\overline{AC}$ at $P$ and $Q$ respectively. Prove that $AP \times AQ \times BD \times DC = AD^2 \times PB \times QC$. From this,prove that $D$ is the midpoint of $\overline{BC}$ if $\overleftrightarrow{PQ} \parallel \overleftrightarrow{BC}$.

In $\Delta XYZ, m\angle Y = 90^{\circ}$. If $XY = YZ$ and $XZ = 10$,then $XY = \ldots$

Prove that if in a triangle the square of one side is equal to the sum of the squares of the other two sides,then the angle opposite the first side is a right angle.