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bisectors of angle A B and C of a triangle ABC intersect its circumcircle at D , E and F respectively . Prove that the angles of the triangle are 90 -1/2 A , 90 – ½ B and 90 -1/2 C.

bisectors of angle A B and C of a triangle ABC intersect its circumcircle at D , E and F respectively . Prove that the angles of  the triangle are 90 -1/2 A , 90 – ½ B and 90 -1/2 C.

Grade:10

1 Answers

Yash Mandil
23 Points
6 years ago
there is also a image of this question which I can not insert in answer that’why
I have asked the same question with image 
kindly refer to that question for image
 
It is given that BE is the bisector of ∠B
.: ∠ABE = ∠B/2   
However, ∠ADE = ∠ABE (Angles in the same segment for chord AE)
  • ∠ADE = ∠B/2
Similarly, ∠ACF = ∠ADF =  ∠C/2 (Angle in the same segment for chord AF) 
 ∠D = ∠ADE + ∠ADF
 ∠D = ∠B/2  + ∠C/2
      = ½  ( ∠B + ∠C )
    = ½  (  180 – ∠A )        {since ∠A + ∠B + ∠C =180 ; angle sum property }
    = 90 – ∠A/2 
Similarly, it can be proved that
∠E = 90 – ∠B/2     &
∠F = 90 – ∠C/2
 

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