Given : △ABC in which a line parallel to side BC intersects other two sides AB and AC at D and E respectively.
To prove : AD/DB = AE/EC
Construction : Join BE and CD and draw DM and EN perpendicular to AC and AB respectively.
Proof : area of △ADE = ½ × base × height
So, ar(ADE) = ½ × AD × EN
Similarly, ar(BDE) = ½ × BD × EN
ar(ADE) = ½ × AE × DM and ar(DEC) = ½ × EC × DM
Note that △BDE and △DEC are on the same base DE and between the same parallels BC and DE.
So, ar(BDE) = ar(DEC)…….(iii)
Therefore, from (i), (ii) and (iii), we have :