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 & Nisha Doshi, (Y10) The Mount School, York.

POQ = 2 x PRQ and SOR = 2 x 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.