In the given figure, if $AB=CD$, then prove that $\mathrm{AC}=\mathrm{BD}$. Also, write Euclid's axiom used for proving it.

In the given figure, if $PQ=QR=RS$ then prove that $PQ=\frac{1}{3}PS$.

In the given figure, if $\mathrm{AB}=\mathrm{AD}$ and $\mathrm{AC}=\mathrm{AD}$, then prove that $\mathrm{AB}=\mathrm{AC}$. State Euclid's axiom to support this.

In the given figure, if $X$ and $Y$ are mid-points of $AB$ and $BC$ and $AX=CY$ then show that $\mathrm{AC}=\mathrm{BC}$.

In the given figure, if $\mathrm{AC}=\mathrm{DC}$ and $\mathrm{CB}=\mathrm{CE}$ then show that $\mathrm{AB}=\mathrm{DE}$.