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Explain theorem 10.10 class 9

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Explain theorem 10.10 class 9
  • 1 answers

Gaurav Seth 3 years, 11 months ago

10.10 theorem :

If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment , the four points lie on a circle i.e (they are concyclic)

 

A , B , C , D  are 4 points ( no 3 of them are collinear)

 

Lets draw a circle passing through A , B , C with Center O.

Assume D does not lie on circle

and circle intersect AD at E

=> A , B , C & E are concyclic

 

now ∠ACB = ∠AEB    ( Angle formed by same chord in same arc segment)

 

∠ACB = ∠ADB  given

 

in Δ DEB

∠AEB = ∠EDB + ∠EBD

=> ∠AEB = ∠ADB + ∠EBD

=> ∠ACB = ∠ACB  + ∠EBD

=> ∠EBD = 0°

 

=> E & D Coincide

 

A , B , C & E are concyclic

E & D coincide

=> A , B , C & D are concyclic

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