In a parallelogram PQRS, M and N are the points on `bar(PQ)` and `bar(RS)` such that PM=RN. prove that MS is parllel to NQ

ABCD is a parallelogram and P, Q are the mid points of bar(AD) and bar(BC) respectively. Show that : bar(BD) and bar(PQ) bisect each other.

ABCD is a parallelogram and P, Q are the mid points of bar(AD) and bar(BC) respectively such that AP=1/3AD and CQ=1/3BC,Show that : AQCP is a parallelogram.

PQRS is a parallelogram. K. L are the midpoints on PQ, RS respectively. bar(PL) intersects bar(SK) at M and RK and QL intersects at N. Show that KNLM is a parallelogram.

ABCD is a parallelogram and P is a point of bar(CD) . Show that ar(/_\ABP) = ar(/_\ADP+ /_\BCP) .

The diagonal AC of a parallelogram ABCD intersects DP at the point Q, where 'P' is any point on side AB. Prove that CQ times PQ=QA times QD .

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).

PQR is a Triangle right angled at P and M is a point on QR such that PMbotQR . Show that PM^2 =QM.MR.

OPQR is a square and M,N are the middle points of sides PQ and QR respectively then the ratio of the area of the square and the triangle OMN is