Prove that `3OD+DA+DB+DC` is equal to OA+OB+OC.

In quadrilateral ABCD, BA = BC and DA = DC. Prove that BD bisects angle ABC as well as angle ADC.

Let O A C B be a parallelogram with O at the origin and O C a diagonal. Let D be the midpoint of O Adot using vector methods prove that B Da n dC O intersect in the same ratio. Determine this ratio.

In the triangle ABC, lines OA,OB and OC are drawn so that angles OAB, OBC and OCA are each equal to omega , prove that cotomega=cotA+cotB+cotC

On the xy plane where O is the origin, given points, A(1, 0), B(0, 1) and C(1, 1) . Let P, Q, and R be moving points on the line OA, OB, OC respectively such that overline(OP)=45t overline((OA)),overline(OQ)=60t overline((OB)),overline(OR)=(1-t) overline((OC)) with t>0. If the three points P,Q and R are collinear then the value of t is equal to

If O is the circumcentre and O' the orthocenter of DeltaABC prove that (i) SA+SB+SC=3SG, where S is any point in the plane of DeltaABC . (ii) OA+OB+OC=OO' Where, AP is diameter of the circumcircle.

If in the given figure, OA=a,OB=b and AP:PB=n:m then OP is equal to

In the triangle OAB , M is the midpoint of AB, C is a point on OM, such that 2 OC = CM . X is a point on the side OB such that OX = 2XB. The line XC is produced to meet OA in Y. Then (OY)/(YA)=

ABCD is a parallelogram. L is a point on BC which divides BC in the ratio 1:2 . AL intersects BD at P.M is a point on DC which divides DC in the ratio 1 : 2 and AM intersects BD in Q. PQ : DB is equal to

Prove that the distances traversed during equal intervals of time by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity [namely 1: 3: 5: …………….].