Question

D and E are points on sides AB and AC respectively of triangle ABC such tht ar(DBC)= ar(EBC).Prove that DE is parallel to BC

Answer 1

given : area(ΔDBC) = area(ΔECB)

construction: 

draw the lines DP and EQ perpendicular to BC from points D and E respectively.

area(ΔDBC) = 1/2*BC*DP

area(ΔECB) = 1/2*BC*EQ

therefore

1/2*BC*DP = 1/2*BC*EQ

i.e. DP = PQ.

since DP and PQ are perpendicular to same line.

therefore DP || PQ

now in the quadrilateral DEQP, DP = PQ

and DP || PQ

one pair of opposite sides are parallel and equal.

therefore DEQP is a parallelogram

and DE || PQ

i.e. DE || BC

Answer 2

Hi friend..
See the attached file

I hope it will help you

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