Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn, intersecting AC in L and AD produced in El. Prove that EL= 2BL

Explanation

Side `AD||` side `BC` ……(Opposite sides of a parallelogram)
i.e. `AE||BC` and `BE` is the transversal.
`:./_BEA~=/_EBC`…….(Alternate angles)
i.e. `/_MED~=/_MBC`
`/_DME~=/_BMC`…..(vertically opposite angles)
`:.DeltaEMD~DeltaBMC`.......("AA" test for similarity)
`:.(MD)/(MC)=(ED)/(BC)`......(Corresponding sides of similar triangles are proportional)
`MD=MC`....(`M` is the midpoint of `DC`)
`:.(MD)/(MC)=(ED)/(BC)=1`
`:.ED=BC` .........`(1)`
`BC=AD`.......(Opposite sides of a parallelogram).....`(2)`
From `(1)` and `(2)`, `ED=AD`
`AD+ED=AD+AD`.......(Adding `AD` to both the sides)
`:.AE=2AD`......`(A-D-E)`....`(3)`
`:.AE=2BC`......(`:'AD=BC`, opposite sides of paraellelogram)
`AE||BC` and `AC` is transversal.
`:./_EAC~=/_BCA`.....(Alternate angle)
`/_ELA~=/_BLC`......(Vertically opposite angles)
`:.DeltaAEL~DeltaCBL`......("AA" test for similarity)
`:.(EL)/(BL)=(AE)/(BC)` (Corresponding sides of similar triangles are proportional)
`:.(EL)/(BL)=(2BC)/(BC)`......[From `(3)`]
`:.(EL)/(BL)=2`
`:.EL=2BL`.