In trapezium ABCD, AB || CD and E is the midpoint of AD. A line drawn through E and parallel to AB intersects BC at F. Prove that F is the midpoint of BC and `EF = (1)/(2)(AB + CD)`.

In Delta ABC , AD is a median and E is the midpoint of AD. BE is extended to intersect AC at F. Prove that AF = (1)/(3)AC .

In triangle ABC, the bisectors of angle B and angle C intersect at I. A line drawn through I and parallel to BC intersects AB at P and AC at Q. Prove that PQ = BP + CQ.

ABC is a triangle right angled at C. A line through the midpoint M of hypotenuse AB and Parallel to BC intersects AC at D. Show that (i) D is the midpoint of AC (ii) MD_|_AC (iii) CM=MA=(1)/(2)AB.

In triangle ABC, AB gt AC and D is any point of BC. Prove that, AB gt AD.

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that DeltaABE ~ DeltaCFB .

In rectangle ABCD, E is the midpoint of side BC. Prove that, AE = DE.

In the adjacent figure ABCD is a parallelogram and E is the midpoint of the side BC. If DE and AB are produced to meet at F, show that AF = 2AB .

In triangleABC , point D lies on side BC. E is the midpoint of AD. Prove that, ar(EBC)= 1/2 ar(ABC)

In triangleABC , AD is a median . E is the midpoint of AD and F is the midpoint of AE. Prove that ar(ABF)= 1/8 ar(ABC).