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Rule

Converse Triangle Proportionality Theorem

If a segment is drawn between two sides of a triangle such that it divides the sides proportionally, the segment is parallel to the third side in the triangle.

Based on the diagram, the following relation holds true.

If AD/DC=BE/EC, then DE ∥ AB.

Proof

The given proportion can be rearranged to get the proportionality of two sides of △ ABC and △ DEC.

AD/DC=BE/EC

▼

Rewrite

AddEqn

LHS+1=RHS+1

AD/DC+1=BE/EC+1

OneToFrac

Rewrite 1 as a/a

AD/DC+DC/DC=BE/EC+EC/EC

AddFrac

Add fractions

AD+DC/DC=BE+EC/EC

Segment Addition Postulate

AC/DC=BC/EC

This means that △ ABC is a dilation of △ DEC from point C with scale factor r = ACDC= BCEC. A dilation moves a segment to a parallel segment, so the proof is complete.

If AD/DC=BE/EC, then DE ∥ AB.

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