Problem

Prove that for each n\ge 4 every convex cyclic quadrilateral can be decomposed into n cyclic quadrilaterals.

Solution

Observe that any triangle can be decomposed into 3 cyclic quadrilaterals by choosing a point in its interior whose pedal points all lie inside the correspondinf sides of the triangle. A triangle can also be decomposed into a triangle and a cyclic quadrilateral. By induction, a triangle can be decomposed into k cyclic quadrilaterals for any k\ge 3.

Any convex quadrilateral can be decomposed into two triangles and, therefore, into k\ge 6 cyclic quadrilaterals. A proof that a cyclic quadrilateral can be decomposed into 4 cyclic quadrilaterals can be found on a separate page. It remains to be shown that a required decomposition exists for k=5.

The result will follow from a sequence of assertions:

  1. A cyclic quadrilateral can be dissected into a trapezoid and a cyclic quadrilateral (by a line parallel to any of its sides).
  2. An isosceles trapezoid is cyclic.
  3. A trapezoid that is not isosceles can be cut into a triangle and an isosceles trapezoid.
  4. A triangle, as we already argued, can be cut into 3 cyclic quadrilaterals.