Question

prove that the angle bisectors of a cyclic quadrilateral from another cyclic quadrilateral

Answer 1

Given: ABCD is a cyclic quadrilateral whose angle bisectors form the quadrilateral PQRS.

To Prove: PQRS is a cyclic

Proof: ABCD is a cyclic quadrilateral ∴∠A +∠C = 180° and ∠B+ ∠D = 180°

      ½ ∠A+½ ∠C = 90° and ½ ∠B+½ ∠D = 90°

    x + z = 90° and y + w = 90°

  In ΔARB and ΔCPD, x+y + ∠ARB = 180° and z+w+ ∠CPD = 180°

  ∠ARB = 180° – (x+y) and ∠CPD = 180° – (z+w)

  ∠ARB+∠CPD = 360° – (x+y+z+w) = 360° – (90+90)

              = 360° – 180°  ∠ARB+∠CPD = 180°

    ∠SRQ+∠QPS = 180°

  The sum of a pair of opposite angles of a quadrilateral PQRS is 180°. Fig

  Hence PQRS is cyclic