$P$ is a point equidistant from two lines $l$ and $m$ intersecting at point $A$ (see Fig.). Show that the line $AP$ bisects the angle between them.

   

$D$ is a point on side $BC$ of $△ABC$ such that $AD=AC$Show that $AB>AD$.

$ABC$ is an isosceles triangle in which $AB=AC$. $AD$ bisects exterior angle $PAC$ and $CD\parallel AB$ (see Fig.). Show that $\angle DAC=\angle BCA$.

$ABC$ is an isosceles triangle in which $AB=AC$. $AD$ bisects exterior angle $PAC$ and $CD\parallel AB$ (see Fig.). Show that $ABCD$ is a parallelogram.

Two parallel lines $l$ and $m$ are intersected by a transversal $p$ (see Fig.). Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.   

$ABCD$ is a parallelogram in which $P$ and $Q$ are mid-points of opposite sides $AB$ and $CD$ (see Fig.). If $AQ$ intersects $DP$ at $S$ and $BQ$ intersects $CP$ at $R$, show that $APCQ$ is a parallelogram, $DPBQ$  is a parallelogram and $PSQR$ is a parallelogram.