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Bisectors of angles A, B and C of a tri...

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`

Text Solution

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It is given that `BE` is the bisector of `angleB, AD` is the bisector of `angleA and CF` is the bisector of `angleC`
. Thus, `angleABE = angleB/2`
However, `angleADE = angleABE`
(Angles in the same segment for chord AE)
Thus, `angleADE = angleB/2`
Similarly, `angleADF = angleACF = angleC/2` (Angle in the same segment for chord AF)
`angleD = angleADE + angleADF`
= `angleB/2 + angleC/2` [Since `angleADE = angleB/2 and angleADF = angleC/2`]
= `1/2 (angleB + angleC )`
= `1/2 (180^@ - angleA)` [Angle sum property of triangle ABC]
= `90^@ - 1/2 A`
Similarly, it can be proved for`angleE = 90^@ - 1/2 B`
`angleF = 90^@ - 1/2 C`
Thus we have proved that the angles of the `triangle DEF` are
`90^@ - 1/2 A, 90^@ - 1/2 B, 90^@ - 1/2 C`.
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