How do you prove that a line parallel to one side of a triangle divides the other two sides proportionally?

Question

nailpolisheverywhere · Answer

• Triangle proportionality theorem also called the side-splitter theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the other two sides proportionally.

anonymous · Answer

thanks @nailpolisheverywhere

nailpolisheverywhere · Answer

you're welcome :)

anonymous · Answer

|dw:1402602311213:dw|

anonymous · Answer

consider 2 triangles: ABC and EBF, we have

anonymous · Answer

1/ $\angle BEF =\angle BAC$ because EF//AC
2/ $\angle BFE=\angle BCA$ the same reason with above
3/ $\angle ABC =\angle EBC$  common angle

anonymous · Answer

therefore, $\triangle ABC $ similar $\triangle EBF$

anonymous · Answer

the properties of 2 similar triangles give us $\dfrac{EB}{AB}=\dfrac{BF}{BC}$ 
And that means EF // AC divides the other 2 sides proportionally.

anonymous · Answer

oh okay. Thank you for explaining @OOOPS

anonymous · Answer

np

anonymous · Answer

can you help me with another questin
 @nailpolisheverywhere