XY is a line parallel to side BC of a triangle AB

Question

XY is a line parallel to side BC of a triangle ABC.If BE | | AC and CF || AB meet XY at Eand F respectively.Show that ar{ABE} = ar{ACF}

Arun · Accepted Answer

Dear Shashwat

Given : A triangle ABC. XY is parallel to BC; BE is parallel to AC and CY is parallel to AB.

To Prove : ar ( ABE ) = ar ( ACF ).

Proof : IIgm EBCY and triangle ABE being on the same base EB and between the same paralllels EB and CA, we have

ar ( ABE ) = 1/2 ar ( IIgm EBCY ) ...................... ( 1 )

Again, IIgm BCFX and triangle ACF being on the same base CF and between the same parallels CF and BA, we have

ar ( ACF ) = 1/2 ar ( llgm BCFX ) ....................... ( 2 )

But, llgm EBCY and llgm BCFX being on the same base BC and between the same parallels BC and EF , we have

ar ( llgm EBCY ) = ar ( llgm BCFX ) ....................... ( 3 )

From, ( 1 ), ( 2 ), and ( 3 ) , we get

ar ( ABE ) = ar ( ACF )

Regards

Arun (askIITians forum expert)

XY is a line parallel to side BC of a triangle ABC.If BE | | AC and CF || AB meet XY at Eand F respectively.Show that ar{ABE} = ar{ACF}