Question

Show that the quadrilateral formed by angle bisectors of a cyclic quadrilateral, is
also cyclic.
​

Answer 1

Answer:

Here, ABCD is a cyclic quadrilateral.

AH,BF,CF and DH are the angle bisectors of ∠A,∠B,∠C and ∠D.

⇒ ∠FEH=∠AEB --- ( 1 ) [ Vertically opposite angles ]

⇒ ∠FGH=∠DGC ---- ( 2 ) [ Vertically opposite angles ]

Adding ( 1 ) and ( 2 ),

⇒ ∠FEH+∠FGH=∠AEB+∠DGC --- ( 3 )

Now, by angle sum property of a triangle,

⇒ ∠AEB=180

o

−(

2

1

∠A+

2

1

∠B) ---- ( 4 )

⇒ ∠DGC=180

o

−(

2

1

∠C+

2

1

∠D) ---- ( 5 )

Substituting ( 4 ) and ( 5 ) in equation ( 3 )

⇒ ∠FEH+∠FGH=180

o

−(

2

1

∠A+

2

1

∠B)+180

o

−(

2

1

∠C+

2

1

∠D)

⇒ ∠FEH+∠FGH=360

o

−

2

1

(∠A+∠B+∠C+∠D)

⇒ ∠FEH+∠FGH=360

o

−

2

1

×360

o

⇒ ∠FEH+∠FGH=180

o

Now, the sum of opposite angles of quadrilateral EFGH is 180