Hello, soni!
There must be a typo . . . The problem makes no sense.
$\displaystyle X\text{ and }Y$ are midpoints of arcs $\displaystyle AB\text{ and }CD$ of a circle
and segment $\displaystyle XY$ cuts chords $\displaystyle AB\text{ and }CD$ at $\displaystyle P\text{ and }Q$, resp.
Prove: .$\displaystyle AP=AQ$ . ?? Code:
C
A * * o
o | *
* | | *
* | | *
| |
* |P |Q *
X o - + - - - + - - - o Y
* | | *
| |
* | | *
* | | *
o | *
B * * o
D
Draw segment $\displaystyle AQ.$
We can see that $\displaystyle AP$ is never equal to $\displaystyle AQ.$