ABCD is a parallelogram. AC and BD are the diagonals intersect at O. P and Q are the points of tri section of the diagonal BD. Prove that `CQ" ||" AP` and also AC bisects PQ (see figure).

In a parallelogram ABCD, E and F are the midpoints of the sides AB and DC respectively. Show that the line segments AF and EC trisect the diagonal BD.

Let A B C D be a parallelogram whose diagonals intersect at P and let O be the origin. Then prove that vec O A+ vec O B+ vec O C+ vec O D=4 vec O Pdot

ABCD is a parallelogram and P is the midpoint of the side bar(BC) . Prove that bar(AC) " and " bar(DP) meet in a common point of trisection.

ABCD is a parallelogram AP and CQ are perpendiculars drawn from vertices A and C on diagonal BD (see figure) show that (i) DeltaAPB ~= DeltaCQD (ii) AP = CQ

In the cyclic quadrilateral ABCD , the diagonal BD bisects the diagonal AC . Prove that ABxxAD=CBxxCD .

P is any point on the diagonal BD of the parallelogram ABCD. Prove that DeltaAPD=DeltaCPD .

ABCD is a parallelogram, P and Q are the mid-points of the sides bar(AB) " and " bar(DC) respectively. Show that bar(DP) " and " bar(BQ) trisect bar(AC) and are trisected by bar(AC) .

ABCD is a parallelogram. The diagonals AC and BD intersect each other at ‘O’. Prove that ar(DeltaAOD) = ar(DeltaBOC) . (Hint: Congruent figures have equal area)

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 -

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. Point Q divides DB in the ratio -