Question

Bisectors of angles A b and c of a triangle ABC intersect its circumcircle at point d e and f respectively prove that the angle of the triangle DEF are 90 degree-A/2, 90 degree-B/2 and 90 degree-C/2. Solve this problem by statement and reason format.
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Answer 1

Answer:

MATHS

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 triangles DEF are 90o−21A, 90o−21B and 90o−21C.

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ANSWER

In △DEF,

∠D=∠EDF

But ∠EDF=∠EDA+∠FDA      ....angle addition property

Now, ∠EDA=∠EBA and ∠FDA=∠FCA      .....Angles inscribed in the same arc

∴∠EDF=∠EBA+∠FCA

=21∠B+21∠C

[Since BE is bisector of ∠B and CF is bisector ∠C]

∴∠D=2∠B+∠C        ....(1)

Similarly, ∠E=2∠C+∠A and ∠F=2∠A+∠B              ...(2)

Now, ∠A+∠B+∠C=180o        ....angle sum