Question

bisectors of angle a,b and c of triangle abc intersect it's circum circle d,e and f respectively. prove that the angles of triangle def are 90-a/2 , 90-b/2 and90-c/2

Answer 1

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