Question

If all sides of a quadrilateral are produced in the same order (clockwise or anticlockwise) .show that the quadrilateral formed by bisectors of exterior angles is cyclic

Answer 1

All sides of a quadrilateral are produced in the same order (clockwise or anticlockwise).

Given:

A cyclic quadrilateral ABCD.

The angle bisectors AR, BR, CP and DP of internal angles A, B, C and D respectively form a quadrilateral PQRS.

To prove:

PQRS is a cyclic quadrilateral.

Proof:

In  Δ ARB,

1/2∠A + 1/2∠B + ∠R = 180° ....(1)

(∵ AR, BR are bisectors of ∠A and ∠B)

In Δ DPC,

1/2∠D + 1/2∠C + ∠P = 180° ....(2)

(∵ DP,CP are bisectors of ∠D and ∠C)

Adding (1) and (2), we get

1/2 ∠ A + 1/2 ∠ B + ∠ R + 1/2 ∠ D + 1/2 ∠ C + ∠ P = 180° + 180°

∠ P + ∠ R = 360° - 1/2 (∠ A + ∠ B + ∠ C + ∠ D)

∠ P + ∠ R = 360° - 1/2 × 360° = 360° - 180°

∴ ∠P + ∠R = 180°

The quadrilateral PQRS is a cyclic quadrilateral, as the sum of a pair of opposite angles of quadrilateral PQRS is 180°.