In triangle ABC, the medians BP and CQ are produced upto points M and N respectively such that `BP=PM and CQ=QN`. Prove that :
A is the mid-point of MN.

In triangle ABC, the medians BP and CQ are produced upto points M and N respectively such that BP = PM and CQ = QN. Prove that: M, A and N are collinear.

In triangle ABC, BE ad CF are median M is a point on BE produced such that BE=EM and N is point on CF produced such that CF=FN. Prove that A is the mid-point of MN

In the parallelogram ABCD ,M is mid-point of AC and X,Y are points on AB and DC respectively such that AX = CY. Prove that : (i) Triangle AXM is congruent to triangle (ii) XMY is a straight line.

In a triangle ABC,D is the mid - point of BC, AD is produced upto E so that DE = AD. Prove that : AB=EC

In triangle ABC, BE ad CF are median M is a point on BE produced such that BE=EM and N is point on CF produced such that CF=FN. Prove that NAM is a straight line

Let the area of a given triangle ABC be Delta . Points A_1, B_1, and C_1 , are the mid points of the sides BC,CA and AB respectively. Point A_2 is the mid point of CA_1 . Lines C_1A_1 and A A_2 meet the median BB_1 points E and D respectively. If Delta_1 be the area of the quadrilateral A_1A_2DE , using vectors or otherwise find the value of Delta_1/Delta

A parallelogram ABCD has P the mid-point of DC and Q a midpoint of AC such that CQ = (1)/(4) AC. PQ produced meets BC at R. Prove that: R is the mid-point of BC

In trapezium ABCD, AB is parallel to DC. P and Q are the mid-points of AD and BC respectively. BP produced meets CD produced at point E. Prove that : PQ is parallel to AB.

In trapezium ABCD, AB is parallel to DC. P and Q are the mid-points of AD and BC respectively. BP product meets CD produced at point E. Prove that : Point P bisects BE,

M and N are mid- point on sides QR and PQ respectively of /_\ PQR , right-anggled at Q. Prove that : PM^(2)+RN^(2)= 5 MN^(2) .