In the given figure, $XY$ is a diameter. Prove that $XY>XZ$.

In the figure, prove that $AH<HC$ and $DC>DH$, where $AD, CE$ are angular bisectors and $AD\perp BC$, $CE\perp AB$.

In the figure given below, $M$ is any point inside the triangle $PQR$. $PM$ is produced to meet $QR$ in $N$. Prove that $\angle QMN>\angle QPN$.

In the figure given below, $M$ is any point inside the triangle $PQR$. $PM$ is produced to meet $QR$ in $N$. Prove that $\angle QMR>\angle QPR$.

In the given figure, if $\angle 1=\angle 2$, prove that $BA>BD$.

In the figure given below, $ABC$ is a triangle in which $AC>AB$ and the bisectors of angles $B$ and $C$ intersect each other at $O$. Prove that $OC>OB$.

In the figure given below, $AB=AC$, then prove that $CD>BD$.

Diagonal $PR$ and $QS$ of a quadrilateral $PQRS$ intersect each other at $O$. Prove that $PQ+QR+RS+SP>PR+QS$.