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Prove that 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
in Mathematics by Platinum (142,760 points) | 805 views

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Lets have △ABC with line DE∥BC

We have to prove that AD/DB=AE/EC

construct or draw  h1 from E perpendicular to AD, and h2 from D perpendicular to AE.

Draw BE and CD.

Area △ADE/Area △BDE =1/2AD.h1/[1/2DB.h1

                                          =AD/DB Area △ADE/Area △CED=1/2AE.h2/[1/2EC.h2]=AE/EC

but     Area △BDE=Area △CED (equal base and height)

hence Area △ADE/Area △BDE = Area △ADE/Area △CED

Therefore AD/DB = AE/EC

Repeating the same method, we can show that:

AD/AB = AE/AC and AB/BD = AC/CE
by Diamond (53,102 points)
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