 Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question says by sectors of angles A, B and C of a triangle ABC intersect its circum circle at D, E and F respectively. Prove that the angles of the triangle D, E, F are 90 degree minus half of A, 90 degree minus half of B and 90 degree minus half of C. Now from the interpretation of the question we have this figure. Now here we see that A, D, B, E and C, F they are the bisectors of angle A, angle B and angle C respectively. Now we have to show here that angles of triangle D, E, F that means this triangle they are 90 degree minus half of A, 90 degree minus half of B and 90 degree minus half of C. So let us see the solution to this question. First of all we say that angle D is same as angle E, D, A plus angle A, D, F that is angle E, D, A plus angle A, D, F. Now this is equal to angle E, B, A plus angle F, C, A. Now we see that we have angle E, D, A is equal to angle E, B, A and angle A, D, F is equal to angle F, C, A because they are angles in the same segment and we know that angles in the same segment of a circle they are equal to each other. So we have angle E, D, A that means this angle is equal to angle E, B, A that is this angle and angle A, D, F that is this angle is equal to angle F, C, A that means this angle. So we can write it as angle D becomes equal to half of angle B plus half of angle C this happens because E, B is the angle by sector of angle B and F, C is the angle by sector of angle C. This is equal to angle B plus angle C divided by 2. Now similarly we can say that angle E is equal to angle C plus angle A divided by 2 and angle F is equal to angle A plus angle B divided by 2. This implies that angle D is equal to 180 degree minus angle A by 2, angle E is equal to 180 degree minus angle B by 2 and angle F is equal to 180 degree minus angle C by 2. Now this happens because we know that angle A plus angle B plus angle C is equal to 180 degree because we know that sum of three angles of a triangle is always equal to 180 degree. Now we had earlier that angle D was equal to angle B plus angle C by 2, angle B plus angle C we can write as 180 degree minus angle A by 2. So we have this similarly we have angle E and we have angle F. This implies angle D can be written as 180 degree divided by 2 is 90 degree minus angle A by 2. So it is 90 degree minus angle A by 2 angle E is equal to 90 degree minus angle B by 2 and angle F is equal to 90 degree minus angle C by 2. So this implies the angles of the triangle D E F are 90 degree minus half of A, 90 degree minus half of B and 90 degree minus half of C. So this was what we were supposed to prove in this question. I hope that you understood the question and enjoyed the session. Have a good day.