# Bisectors of angles A,B,C of a triangle ABC intersects its circumcircle at D,E, and F respectively. Prove that the angles of triangle DSF are 90 degrees - A/2, 90 degrees -B /2 and 90 degrees -C/2

1
by anudeepd2

Log in to add a comment

by anudeepd2

Log in to add a comment

The Brainliest Answer!

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

See diagram.

This problem can be solved by two methods. One is by using fact that the angles subtended by a chord (or part of the circle) at any point on the circumference of the circle is same. Second is by using the angle subtended at the center by a chord is twice the angle subtended at a point on the circumference.

Here let us the see the solution by the 1st method.

Mark angles BAD = A/2 = angle DAC; angle ACF = angle FCB;

angle ABE = CBE = B/2

Chord FA subtends C/2 at C => FDA = C/2

Chord AE subtends B/2 at B => ADE = B/2

angle FDE = (B+C)/2 = (180-A)/2 = 90 - A/2

Chord BF subtends C/2 at C, so subtends same C/2 at E.

Chord BD subtends A/2 at A, so subtends A/2 at E.

angle FED =( A+C)/2 = (180-B)/2 = 90 -B/2

similarly the other one.

This problem can be solved by two methods. One is by using fact that the angles subtended by a chord (or part of the circle) at any point on the circumference of the circle is same. Second is by using the angle subtended at the center by a chord is twice the angle subtended at a point on the circumference.

Here let us the see the solution by the 1st method.

Mark angles BAD = A/2 = angle DAC; angle ACF = angle FCB;

angle ABE = CBE = B/2

Chord FA subtends C/2 at C => FDA = C/2

Chord AE subtends B/2 at B => ADE = B/2

angle FDE = (B+C)/2 = (180-A)/2 = 90 - A/2

Chord BF subtends C/2 at C, so subtends same C/2 at E.

Chord BD subtends A/2 at A, so subtends A/2 at E.

angle FED =( A+C)/2 = (180-B)/2 = 90 -B/2

similarly the other one.