In the figure,if $OX = \frac{1}{2} XY$,$PX = \frac{1}{2} XZ$ and $OX = PX$,show that $XY = XZ$.

(N/A) Given: $OX = \frac{1}{2} XY$,$PX = \frac{1}{2} XZ$ and $OX = PX$.
Since $OX = PX$,we can substitute the given values:
$\frac{1}{2} XY = \frac{1}{2} XZ$
According to Euclid's axiom,"Things which are double of the same things are equal to one another."
Multiplying both sides by $2$,we get:
$2 \times (\frac{1}{2} XY) = 2 \times (\frac{1}{2} XZ)$
$XY = XZ$
Hence,it is shown that $XY = XZ$.