Prove that the quadrilateral formed by internal angle bisectors of a cyclic quadrilateral is also cyclic.
Given that: ABCD is a cyclic quadrilateral
To Prove: EFGH is also cyclic quadrilateral.
ABCD is a cyclic quadrilateral and we know that the sum of the opposite angles of a cyclic quadrilateral is.
Adding equation (2) and (3), we have,
Substitute value of from equation (1), in above equation, we have,
The sum of opposite angles of a quadrilateral is.
Hence, EFGH is a cyclic quadrilateral.
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