Two perpendicular chords of a circle meet at a point inside the circle and cut off arcs $a$, $b$, $c$ and $d$ on the circumference of the circle. What is the relationship between the arcs $a$, $b$, $c$ and $d$?

Here is another excellent solution from Suzanne Abbott and Nisha Doshi, (Y10) The Mount School, York.

$POQ = 2 \times PRQ$ and $SOR = 2 \times SPR$.

But since $PRQ + SPR = 90$ then it follows that $POQ + SOR = 180$.

Also since the length of the arcs are directly related to the angles at the centre of the circle it follows that $a + c$ is a half of the circumference. So $a + c = b + d$.