Question

Theorem:The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.
❌No spam ❌
Explain ​

Answer 1

$\huge{\underline{\red{\mathcal{AnsWer}}}}$

Refer attachment..

➡➡➡➡➡➡➡➡➡➡➡➡➡➡➡➡

Answer 2

$\underline \mathbb{SOLUTION}$

$\bold{\fbox{\color{Red}{Given,}}}$

》A cyclic quadrilateral ABCD in which AP, BP, CR and DR are the bisectors of Angle A , Angle B , angle C and angle D respectively forming a quadrilateral PQRS.

$\bold{\fbox{\color{Red}{To\:Prove}}}$

》PQRS is a cyclic quadrilateral.

$\bold{\fbox{\color{Red}{Proof}}}$

$\implies\:$IN ΔPAB,

∠APB + ∠PAB + ∠PBA = 180°

[sum of the angles of ΔPAB = 180°]

$\implies\:$ ∠APB + 1/2 ∠A + 1/2 ∠B = 180°————(1)

[∠PAB = 1/2∠A and ∠PBA = 1/2∠B]

$\implies\:$ IN ΔRCD,

∠CRD + ∠RCD + ∠RDC =180°

[sum of the angles of ΔRCD= 180°]

$\implies\:$ ∠CRD + 1/2 ∠C + ∠1/2 ∠D = 180°——— (2)

[∠RCD = 1/2∠C and ∠RCD =1/2 ∠D]

∠APB + ∠CRD + 1/2 (∠A + ∠B + ∠C + ∠D) = 360°

[ADDING (1) and (2)]

$\implies\:$∠APB + ∠CRD +1/2 × 360° =360°

∠A + ∠B + ∠C + ∠D = 360°

$\implies\:$ ∠APB + ∠CRD = 180°

》Sum of pair of opposite angles of quadrilateral PQRS is 180°

》PQRS is a cyclic quadrilateral.

$\bold\green{\underline{Hence\:proved}}$

_______________________________________

NOTE:

REFER TO ATTACHMENT FOR FIGURE.

_______________________________________