Draw circle centre O.
Draw chord AB, dividing the circle into two sectors, a major sector and a minor sector. In each sector mark a point, C and D. The four points A, B, C and D can be joined to form a quadrilateral ACBD.
Join O to A and B. The smaller angle at the centre AOB subtends an angle at the circumference of the major sector equal to half the angle at the centre; the larger angle at the centre AOB subtends an angle at the circumference of the minor sector equal to half the angle at the centre. Since the sum of the smaller and larger angles AOB is 360 degrees, then the angles at the circumference must be complementary, i.e., they add up to 180 degrees. These are the opposite interior angles of the quadrilateral. ACB+ADB=180.
By extending the opposite sides AD and BC of the quadrilateral, we can see the exterior angles of these sides, and we know that the exterior and interior angles are complementary because they lie on a straight line. The exterior angles at vertices C and D therefore respectively equal the interior angles at D and C, so their bisectors will be half these values. Since they are complementary, half their sum must be 90 degrees, so they meet at right angles.