Question

The quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.

• A. True
• B. False

Answer 1

The correct option is A True

In the given figure $ABCD$ is a cyclic quadrilateral.

$AH,BF,CF$ and $DH$ are the angle bisectors of $\angle A,\angle B,\angle C$ and $\angle D$

$\Rightarrow$ $\angle FEH=\angle AEB$ [ Vertically opposite angles ] ---- ( 1 )

$\Rightarrow$ $\angle FGH=\angle DGC$ [ Vertically opposite angles ] ---- ( 2 )

$\Rightarrow$ $\angle FEH+\angle FGH=\angle AEB+\angle DGC$ [ Adding ( 1 ) and ( 2 ) ]

Now, by angle sum property of a triangle,

$\angle AEB={180}^o-(\frac{1}{2}\angle A+\frac{1}{2}\angle B)$ and

$\angle DGC={180}^o-(\frac{1}{2}\angle C+\frac{1}{2}\angle D)$

$\Rightarrow$ $\angle FEH+\angle FGH={180}^o-(\frac{1}{2}\angle A+\frac{1}{2}\angle B)+{180}^o-(\frac{1}{2}\angle C+\frac{1}{2}\angle D)$

$\Rightarrow$ $\angle FEH+\angle FGH={360}^o-\frac{1}{2}(\angle A+\angle B+\angle C+\angle D)$

$\Rightarrow$ $\angle FEH+\angle FGH={360}^o-\frac{1}{2}\times {360}^o$

$\Rightarrow$ $\angle FEH+\angle FGH={180}^o$

Now, the sum of opposite angles of quadrilateral $EFGH$ is ${180}^o.$

$\therefore$ $EFGH$ is a cyclic quadrilateral.

Hence, the quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.

That is, the given statement is true and option $A$ is correct.