`If a line is drawn to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.

Explanation

Let a `DeltaABC` in which a line DE parallel to BC intersects AB to D and AC at E.
To prove DEE divides the two sides in the same ratio.
i.e., `(AD)/(DB)=(AE)/(EC)`

Construction Join BE, CD and draw EF`bot`and DG`bot`AC.
Proof Here `(ar(DeltaADE))/(ar(DeltaBDE))=(1/2xxADxxEF)/(1/2xxDBxxEF)` [`because`area of triangle=`1/2xx`base`xx`height]
`=(AD/DB)`...(i)
similarly, `(ar(DeltaADE))/(ar(DeltaDEC))=(1/2xxAExxGD)/(1/2xxECxxGD)=(AE)/(EC)`....(ii)
Now since, `DeltaBDE and DeltaDEC` lie between the same parallel DE and BC and on the same base DE.
So, `ar(DeltaBDE)=ar(DeltaDEC)...(iii)`
From Eqs. (i), (ii) and (iii), `(AD)/(DB)=(AE)/(EC)` Hence proved