Answer:
The region between a chord and either of its arcs is called a segment the circle.
Angles in the same segment of a circle are equal.
Step-by-step explanation:
Let bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F .
Now from figure,
∠D = ∠EDF
∠D = ∠EDA + ∠ADF
Since ∠EDA and ∠EBA are the angles in the same segment of the circle.
So ∠EDA = ∠EBA
Hence ∠D = ∠EBA + ∠FCA
Again ∠ADF and ∠FCA are the angles in the same segment of the circle.
hence ∠ADF = ∠FCA
Again since BE is the internal bisector of ∠B and CF is the internal bisector of ∠C
So ∠D = ∠B/2 + ∠C/2
Similarly
∠E = ∠C/2 + ∠A/2
and
∠F = ∠A/2 + ∠B/2
Now ∠D = ∠B/2 + ∠C/2
=∠D = (180 - ∠A)/2
(∠A + ∠B + ∠C = 180)
∠D = 90 - ∠A/2
∠E = (180 - ∠B)/2
∠E = 90 - ∠B/2
and ∠F = (180 - ∠C)/2
∠F = 90 - ∠C/2
