prove Thales theorem....
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: If a line is drawn parallel to one side of a triangle, to interest the other two sides at indistinct points, the other two sides are divided in the same ratio.
Given: - In △ABC,DE∥BC
To prove:- AD/DB = AE/EC
Construction:- BE and CD are joined. EF⊥AB and DN⊥AL
are drawn.
Proof:-
ar(△ADE) = 1/2×AD×EF
=1/2×AE×DN
ar(△BDE) = 1/2×BD×EF
ar(△CDE) = 1/2×EC×DN
ar(△ADE)/ar(△BDE) = 1/2×AD×EF/1/2×BD×EF ..(1)
ar(△ADE)/ar(△CDE) = 1/2×AE×DN/1/2×EC×DN (2)
But, ar(△BDE) = ar(△CDE) (3)
as they are on the same base DE and DE∥BC
from (1), (2) and (3) we get AD/DB = AE/EC
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