Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are `90^@-1/2A, 90^@-1/2B" and "90^@-1/2C`

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#### Solution

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

`=(angleB)/2 + (angleC)/2`

`=1/2(angleB+angleC)`

`=1/2(180^@-angleA)`

`=90^@-1/2angleA`

Similarly, it can be proved that

`angleE=90^@-1/2angleB`

`angleF=90^@-1/2angleC`

Concept: Cyclic Quadrilateral

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