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XY is a line parallel to side BC of a t...

XY is a line parallel to side BC of a triangle ABC. If `B E\ ||\ A C`and `C F\ ||\ A B`meet XY at E and F respectively, show that `a r\ (A B E)\ =\ a r\ (A C F)`

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Since, `BE||AC rArr BE ||YC`...(1)
and `BC || EY` (given)...(2)
`:. squareBCYE` is a parallelogram (`:.` pairs of opposite sides are parallel)
Now, since `DeltaABE` and parallelogram BCYE are on the same
base EB and between the same parallels EB and AC,
`:. ar(DeltaABE) = (1)/(2) ar(||gm BCYE)`...(3)
Also, since `CF ||AB rArr CF ||BX`
and `BC||XF` (given)
`:. square BCFX` is a paralleogram (`:.` pairs of opposite sides are parallel)
Now, since `DeltaACF` and parallelogram BCFX are on the same base CF and between the same parallels CF and AB,
`:. ar(DeltaACF) = (1)/(2) ar(||gm BCFX)`....(4)
But `ar(||gm BCYE) = ar(||gm BCFX)`...(5)
(`:.` parallelograms are on the same base BC and between same parallels BC and EF)
`:.` From (3), (4) and (5), we have
`ar(DeltaABE) = ar(DeltaACF)`
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