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triangle proportionality theorem — Definition, Formula & Examples

The Triangle Proportionality Theorem states that if a line runs parallel to one side of a triangle and intersects the other two sides, it splits those two sides into segments that are proportional. It is also called the side-splitter theorem.

In triangle ABCABC, if a line ℓ\ell is parallel to side BC‾\overline{BC} and intersects AB‾\overline{AB} at point DD and AC‾\overline{AC} at point EE, then ADDB=AEEC\dfrac{AD}{DB} = \dfrac{AE}{EC}.

Key Formula

ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}
Where:
  • ADAD = Length of the segment from vertex A to point D on side AB
  • DBDB = Length of the segment from point D to vertex B on side AB
  • AEAE = Length of the segment from vertex A to point E on side AC
  • ECEC = Length of the segment from point E to vertex C on side AC

How It Works

Draw any triangle and then draw a line inside it that is parallel to one of its sides, crossing the other two sides. The theorem guarantees that the two sides are cut in the same ratio. If one side is split into segments of lengths 3 and 6, the other side must also be split in the ratio 3:63:6 (which simplifies to 1:21:2). You can use this to find an unknown segment length by setting up a proportion and solving for the missing value. The converse is also true: if a line divides two sides of a triangle proportionally, then that line must be parallel to the third side.

Worked Example

Problem: In triangle ABC, line DE is parallel to BC, with D on AB and E on AC. If AD = 4, DB = 6, and AE = 5, find EC.
Set up the proportion: By the Triangle Proportionality Theorem, the parallel line creates proportional segments.
ADDB=AEEC  ⟹  46=5EC\frac{AD}{DB} = \frac{AE}{EC} \implies \frac{4}{6} = \frac{5}{EC}
Cross-multiply and solve: Cross-multiply to isolate EC.
4⋅EC=6⋅5=30  ⟹  EC=304=7.54 \cdot EC = 6 \cdot 5 = 30 \implies EC = \frac{30}{4} = 7.5
Answer: EC=7.5EC = 7.5

Why It Matters

This theorem appears repeatedly in high school geometry proofs and standardized tests whenever parallel lines cut through triangles. Architects and surveyors rely on it to calculate inaccessible distances by measuring proportional segments in similar triangles formed by parallel reference lines.

Common Mistakes

Mistake: Setting up the proportion using the full side length instead of the two segments. For example, writing AD/AB = AE/EC.
Correction: Each ratio must compare the two segments on the same side: AD/DB = AE/EC. If you use a full side like AB, the other term must also be a full side (AD/AB = AE/AC), but both ratios must be structured consistently.

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