Thales Theorem

Statement


"If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio "
Given
To Prove that
Construction
Join B to E and C to D. Draw perpendicular EN on AB and DM on AC.
Proof


ABC is a triangle. DE parallel to BC.

AD/DB = AE/EC

Consider in triangle ADE and BDE.
ar(ADE) = 1/2 ADxEN
ar(BDE) = 1/2DBxEN
Therefore,
ar(ADE)/ar(BDE) = AD/DB ......... (i)
Again, consider in triangle ADE and CDE.
ar(ADE) = 1/2 AExDM
ar(CDE) = 1/2 ECxDM
Therefore,
ar(ADE)/ar(CDE) = AE/EC ............ (ii)
Triangles BDE and CDE have same base DE and DE parallel to BC.
Therefore,
ar(BDE) = ar(CDE)
Now from (i) and (ii)
AD/DB = AE/EC Proved.