The side AC of a triangle ABC is produced to point E so that `CE=(1)/(2)` AC. D is the mid-point of BC and ED produced meets AB at F. Lines through D and C are drawn parallel to AB which meet AC at point P and EF at point R respectively. Prove that:
`3DF=EF`

The side AC of a triangle ABC is produced to point E so that CE=(1)/(2) AC. D is the mid-point of BC and ED produced meets AB at F. Lines through D and C are drawn parallel to AB which meet AC at point P and EF at point R respectively. Prove that: 4CR=AB

D and F are the mid-points of sides AB and AC of a triangle ABC. A line through F and parallel to AB meets BC at point E. Find AB, if EF=4.8cm .

ABC is a triangle A line is drawn parallel to BC to meet AB and AC in D and E respectively. Prove that the median through A bisects DE.

In parallelogram ABCD, E is mid-point of CD and through D, a line is drawn parallel to EB to meet CB produced at point G and to cut AB at point F. Prove that : DG = 2EB

D and F are the mid-points of sides AB and AC of a triangle ABC. A line through F and parallel to AB meets BC at point E. Prove that BDFE is a parallelogram

In a triangle ABC, AD is a median and E is mid-point of median AD. A line through B and E meets AC at point E. Prove that : AC = 3AF Draw DG parallel to BF, which meets AC at point G.

In triangle ABC, P is the mid-point of side BC. A line through P and parallel to CA meets AB at point Q and a line through Q and parallel to BC meets median AP at point R. Prove that : AP=2AR

In parallelogram ABCD, E is mid-point of CD and through D, a line is drawr parallel to EB to meet CB produced at point G and to cut AB at point F. Prove that: 2 xx AD = GC

In /_\A B C ,D is the mid-point of BC and ED is the bisector of the /_ADB and EF is drawn parallel to B C cutting A C in F . Prove that /_E D F is a right angle.

In triangle ABC, D and E are points on side AB such that AD = DE = EB. Through D and E, lines are drawn parallel to BC which meet side AC at points F and G respectively. Through F and Glines are drawn parallel to AB which meet side BC at points M and N respectively. Prove that : BM = MN = NC.