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 circumcircle at D,E and F respectively. Prove that A DcosA/2+B EcosB/2+C FcosC/2=2R(sinA+sinB+sinC)

In an isosceles right-angled triangle ABC, /_B=90^(@) . The bisector of /_BAC intersects the side BC at the point D. Prove that CD^(2) =2BD^(2)

The bisector of the interior angles angleB and angle C of the DeltaABC , meet at H. Prove that angleBHC=90^@+(1)/(2)angleBAC .

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 a and b 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)

Three vertices of triangle ABC are A (3 , 2 , -1) B ( - 1 , -1 , -1) and C ( 1, 5, 5) , if the internal bisector of angleBAC meets the opposite side overline(BC) at D , then find the coordinates of D .

AB is a diameter of a circle. BP is a tangent to the circle at B . A straight line passing through A intersects BP at C and the circle at D . Prove that BC^(2)=ACxxCD .

ABCD is a cyclic quadrilateral .The side BC is extended to E .The bisectors of the angles angleBADandangleDCE intersect at the point P .Prove that angleADC=angleAPC .

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)/triangle

In A B C , internal angle bisector of /_A meets side B C in D . DE_|_A D meets A C at E and A B at F. Then (a) A E is in H.P. of b and c (b) A D=(2b c)/(b+c)cosA/2 (c) E F=(4b c)/(b+c)sinA/2 (d) AEF is isosceles