# 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.

Proof:

ABCD is a cyclic quadrilateral and we know that the sum of the opposite angles of a cyclic quadrilateral is .

So,  Now, …… (1)

Now,

In  …… (2)

And …… (3)

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.

• 150 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

• 42

Thanks.

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