Prove that , "If a line parallel to a side of a triangle intersects the remaining sides in two distinct points then the line divides the sides in the same proportion".

Explanation

Proof `:` `Delta XAB ` and `Delta BAY ` have a common vertex B and their bases XA and AY lie on the same line XY.
`:.` they have equal heights.
`:. (A (Delta XAB))/(A ( Delta BAY )) = ( XA )/( AY ) ` …[Triangles of equal heights ] ….(1)
`Delta XAB ` and `Delta ABZ ` have a common verte A and their bases XB and BZ lie on the same line XZ.
`:.` they have equal heights.
`:. (A( Delta XAB))/( A ( Delta ABZ)) = ( XB)/( BZ)` ....[Triangles of equal heights] ....(2)
`:. Delta BAY ` and `Delta ABZ ` lie between the same two parallel lines AB and YZ.
`:. ` they have equal heights, also have same base AB.
`:. A ( Delta BAY ) = A ( Delta ABZ ) `.... [Triangles of equal bases and heights ] ....(3)
`:.` from (1) , (2) and (3) , we get
`(A(DeltaXAB))/(A(DeltaBAY)) = ( A(Delta XAB))/(A(DeltaABZ))` ....(4)
`:.` from (1) ,(2) and (4) , we get
`( XA)/( AY) = (XB)/( BZ)`