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Bisectors of a and c of a cyclic quadrilateralabcd intersect a circle through a,b,c,d at e and f respectively.proove that ef is the diameter of the circle

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Agam answered 5 month(s) ago

the bisectors of the opp. angles a and c of cyclic quadrilateral abcd intersect the circle at the points e and f respectively prove that ef is a diameter of the circle

The bisectors of the opp. angles A and C of cyclic quadrilateral ABCD intersect the circle at the points E and F respectively. Prove that EF is a diameter of the circle

Class-IX Maths

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Asked by Kanishka

Feb 5

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Ramesh , SubjectMatterExpert

Member since Apr 01 2014

Given: ABCD is a cyclic quadrilateral. AE and CF are the bisectors of ∠A and ∠C respectively.
 
To prove: EF is the diameter of the circle i.e. ∠EAF = 90°
 
Construction: Join AE and FD.
 
Proof: ABCD is a cyclic quadrilateral.
 
∴ ∠A + ∠C = 180° (Opposite angles of a cyclic quadrilateral are supplementary )
 
⇒ ∠EAD + ∠DCF = 90° ------- (1) (AE and CF are the bisector of ∠A and ∠C respectively)
 
∠DCF = ∠DAF -------- (2) (Angles in the same segment)
 
From equations (1) and (2), we get,
 
∠EAD + ∠DAF = 90°
 
⇒ ∠EAF = 90°
 
But ∠EAF is the angle in a semi-circle.
 
∴ EF is the diameter of the circle.

I hope this help ! u