P and Q are any two points lying on the sides DC and AD respectively of a parallelogram ABCD show that `ar(DeltaAPB) = ar Delta(BQC)`

In the figure ''P'' is a point in the interior of a parallelogram ABCD. Show that area of DeltaAPD+ area of DeltaPBC = area of DeltaAPB+ area of DeltaPCD .

P and Q are the mid-points of the sides CA and CB respectively of a Delta ABC , right angled at C. Prove that 4(AQ^(2)+BP^(2))=5 AB^(2) .

If D,E and F are the mid-points of the sides BC, CA and AB respectively of a triangle ABC and lambda is scalar, such that vec(AD) + 2/3vec(BE)+1/3vec(CF)=lambdavec(AC) , then lambda is equal to

In trapezlum ABCD, AB || DC, E and F are points on non-parallel sides AD and BC respectively, such that EF || AB. Show that (AE)/(ED)=(BF)/(FC) .

In trapezium ABCD, AB || DC. E and F are points on non-parallel sides AD and BC respectively such that EF || AB. Show that (AE)/(ED) = (BF)/(FC)

In the figure, ABC is a triangle in which AB= AC. Points D and E are points on the side AB and AC respectively such that AD=AE. Show that points B, C, E and D lie on a same circle.

O P Q R is a square and M ,N are the midpoints of the sides P Q and Q R , respectively. If the ratio of the area of the square to that of triangle O M N is lambda:6, then lambda/4 is equal to (a)2 (b) 4 (c) 2 (d) 16

The points A(-3,6), B(0,7) and C(1,9) are the mid points of the sides DE, EF and FD of a triangle DEF. Show that the quadrilateral ABCD is a parallellogram.

O P Q R is a square and M ,N are the middle points of the sides P Qa n dQ R , respectively. Then the ratio of the area of the square to that of triangle O M N is (a)4:1 (b) 2:1 (c) 8:3 (d) 7:3