The equilateral triangle ABC is inscribed in a circle. If P is any point on the ar BC, then prove that `AB=PB+PC`.

The isosceles triangle ABC is inscribed in a circle with centre at O. If AP is a diameter of the circle. Then prove that AP is the bisetors of angle BPC .

If P(A//B)=1 then prove that, P(ABC)=P(BC) .

Answer any One question : If a quadrilateral ABCD is circumscribed about a circle with centre O, prove that AB + CD = BC + DA.

On the outside of the equilateral triangle ABC and in the angular region angleABC , O is a point. OP, OQ and OR are the perpendiculars drawn from O to the sides. AB, BC and CA respectively. Prove that the height of the triangle = OP + OQ - OR.

The triangle ABCis inscribed in a circle, AX, BY and CZ of the angles angle BAC, angle ABC and angle ACB , intersect at the point X,Y,Z on the circle respectively. Prove that AX is perpendicular to XZ.

Find the area of an equilateral triangle inscribed in the circle x^2 + y^2 + 2gx + 2fy + c = 0

Delta ABC is an equilateral triangle . D is a point on the side BC such that BD = (1)/(3) BC . Prove that 7 AB^(2) = 9AD^(2)

D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) respectively of the triangle ABC. If P is any point in the plane of the triangle, show that vec(PA)+vec(PB)+vec(PC)=vec(PD)+vec(PE)+vec(PF) .

D,E,F are the midpoints of the sides bar(BC),bar(CA) and bar(AB) respectively of the triangle ABC. If P is any point in the plane of the traingle , show that bar(PA)+bar(PB)+bar(PC)= bar(PD)+bar(PE)+bar(PF) .