Q : 8       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    \small 90^{\circ}-\frac{1}{2}C,  \small 90^{\circ}-\frac{1}{2}B and \small 90^{\circ}-\frac{1}{2}A

Answers (1)

Given :   Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively.

To prove :  the angles of the triangle DEF are    \small 90^{\circ}-\frac{1}{2}C,    \small 90^{\circ}-\frac{1}{2}B and \small 90^{\circ}-\frac{1}{2}A

Proof : 

          

\angle1 and \angle3 are angles in same segment.therefore,

          \angle1 = \angle3 ................1(angles in same segment are equal )

and    \angle2 = \angle4 ..................2

Adding 1 and 2,we have 

         \angle1+\angle2=\angle3+\angle4

\Rightarrow \angle D=\frac{1}{2}\angle B+\frac{1}{2}\angle C,

\Rightarrow \angle D=\frac{1}{2}(\angle B+\angle C)

\Rightarrow \angle D=\frac{1}{2}(180 \degree+\angle C)

and  \Rightarrow \angle D=\frac{1}{2}(180 \degree-\angle A)

        \Rightarrow \angle D=90 \degree-\frac{1}{2}\angle A

Similarly,  \Rightarrow \angle E=90 \degree-\frac{1}{2}\angle B     and    \angle F=90 \degree-\frac{1}{2}\angle C

 

 

 

 

 

 

 

 

 

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