In ` A B C ,` the bisector of the angle A meets the side BC at D andthe circumscribed circle at E. Prove that `D E=(a^2secA/2)/(2(b+c))`

In A B C , the three bisectors of the angle A, B and C are extended to intersect the circumeircle at D,E and F respectively. Prove that A DcosA/2+B EcosB/2+C FcosC/2=2R(sinA+sinB+sinC)

The internal bisector of angle A of Delta ABC meets BC at D and the external bisector of angle A meets BC produced at E. Prove that (BD)/(BE) = (CD)/(CE) .

In a cyclic quadrilateral ABCD, the bisectors of opposite angles A and C meet the circle at Pand Q respectively. Prove that PQ is a diameter of the circle.

In convex quadrilateral A B C D ,A B=a ,B C=b ,C D=c ,D A=d . This quadrilateral is such that a circle can be inscribed in it and a circle can also be circumscribed about it. Prove that tan^2A/2=(b c)/(a d)dot

Let aa n db represent the lengths of a right triangles legs. If d is the diameter of a circle inscribed into the triangle, and D is the diameter of a circle circumscribed on the triangle, the d+D equals. (a) a+b (b) 2(a+b) (c) 1/2(a+b) (d) sqrt(a^2+b^2)

In triangle ABC,angleC=(2pi)/3 and CD is the internal angle bisector of angleC meeting the side AB at D. If Length CD is 1, then H.M. of a and b is equal to : (a) 1 (b) 2 (c) 3 (d) 4

A triangle A B C is inscribed in a circle with centre at O , The lines A O ,B Oa n dC O meet the opposite sides at D , E ,a n dF , respectively. Prove that 1/(A D)+1/(B E)+1/(C F)=(acosA+bcosB+ccosC)/

In A B C , internal angle bisector of /_A meets side B C in DdotD E_|_A D meets A C in Ea n dA B in Fdot Then A E is HdotPdot of b and c A D=(2b c)/(b+c)cosA/2 E F=(4b c)/(b+c)sinA/2 (d) A E Fi si sos c e l e s

In a right angled DeltaABC,angleACB=90^(@). A circle is inscribed in the triangle with radius r, a, b, c are the lengths of the sides BC, AC and AB respectively. Prove that 2r=a+b-c.

In the adjacent figure, ABC is a right angled triangle with right angle at B and points D, E trisect BC. Prove that 8AE^(2)=3AC^(2)+5AD^(2).