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.

  • -18
What are you looking for?