Basic Proportionality Theorem states that, if a line is parallel to a side of a triangle which intersects the other sides into two distinct points,then the line divides those sides of the triangle in proportion.
Let ABC be the triangle.
The line l parallel to BC intersect AB at D and AC at E.
To prove DBAD=ECAE
Join BE,CD
Draw EF⊥AB, DG⊥CA
Since EF⊥AB,
EF is the height of triangles ADE and DBE
Area of △ADE=21× base × height=21AD×EF
Area of △DBE=21×DB×EF
areaofΔDBEareaofΔADE=21×DB×EF21×AD×EF=DBAD ........(1)
Similarly,
areaofΔDCEareaofΔADE=21×EC×DG21×AE×DG=ECAE ......(2)
But ΔDBE and ΔDCE are the same base DE and between the same parallel straight line BC and DE.
Area of ΔDBE= area of ΔDCE ....(3)
From (1), (2) and (3), we have
DBAD=ECAE
Hence proved.
