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Prove that if a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
in General Maths by Diamond (40.2k points) | 12 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

Therefore since there is equal ratio on divided sides, the line is parallel to the third side of a triangle.
by Silver Status (31.3k points)

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