Triangle Proportionality Theorem Calculator

Enter three known segment lengths from a triangle split by a line parallel to one side, and solve the fourth using AD/DB = AE/EC.

Quick Facts

Triangle Proportionality Theorem
AD/DB = AE/EC
Holds whenever DE is drawn parallel to BC, with D on AB and E on AC.
Cross-multiplied form
AD × EC = DB × AE
Rearrange this to isolate any one segment from the other three.
Converse of the theorem
Equal ratios ⇒ DE ∥ BC
If AD/DB = AE/EC, the line DE must be parallel to BC.
Whole-side equivalent
AD/AB = AE/AC
The same proportion, written using the full sides instead of the split segments.

Your Results

Calculated
Computed Segment
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Solved from AD/DB = AE/EC
AD : DB Ratio
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AD divided by DB
AE : EC Ratio
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AE divided by EC (should match AD:DB)
Full Side Lengths
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AB = AD + DB, AC = AE + EC

Ready

Enter three known segment lengths and choose which one to solve for.

Formula and Method for the Triangle Proportionality Theorem

The Triangle Proportionality Theorem (also known as the Side-Splitter Theorem) states: if a line segment DE is drawn inside triangle ABC parallel to side BC, intersecting AB at point D and AC at point E, then it divides the two sides it crosses into proportional segments: AD/DB = AE/EC. Given any three of the four segment lengths, the fourth can always be found by solving this proportion — that is exactly what this calculator does.

How the calculation works

Cross-multiplying the proportion AD/DB = AE/EC gives AD × EC = DB × AE. Whichever segment you mark as unknown, the calculator isolates it algebraically: EC = (DB × AE) / AD, AD = (DB × AE) / EC, DB = (AD × EC) / AE, or AE = (AD × EC) / DB. Enter the other three segment lengths in the same unit, choose the unit, and press Calculate. The tool also reports the AD:DB and AE:EC ratios (which will always match once the missing segment is solved) and the full side lengths AB = AD + DB and AC = AE + EC.

Common mistakes

  • Mixing up direction: AD is measured from vertex A to point D, while DB is measured from D to vertex B — swapping them inverts the ratio and gives a wrong answer.
  • Assuming any crossing line qualifies: the proportion AD/DB = AE/EC only holds when DE is parallel to BC. If the line is not parallel, the segments will not be proportional.
  • Mixing segment ratios with whole-side ratios: AD/DB = AE/EC uses the two split segments; AD/AB = AE/AC uses whole sides. Both are valid forms of the theorem, but don't combine a segment from one form with a side from the other in the same equation.

Real-world applications

  • Geometry and trigonometry coursework, where proving or applying similar-triangle relationships is a core skill
  • Scale drawings, architectural plans, and map-making, where a parallel reference line lets you infer an unmeasured distance
  • Construction and carpentry, for checking that a cross-brace, shelf, or cut line is truly parallel to a triangular frame's edge
  • Surveying and indirect measurement, where inaccessible distances are inferred from proportional, measurable ones

Frequently Asked Questions

What is the Triangle Proportionality Theorem?
If a line segment DE is drawn parallel to side BC of triangle ABC, intersecting AB at D and AC at E, then it divides those two sides proportionally: AD/DB = AE/EC. This is also called the Side-Splitter Theorem.
How do I solve for a missing segment with this theorem?
Cross-multiply the proportion AD/DB = AE/EC to get AD × EC = DB × AE, then divide by whichever segment is known on the same side as the missing one. For example, if EC is unknown, EC = (DB × AE) / AD.
What is the converse of the Triangle Proportionality Theorem?
The converse states that if a line intersects two sides of a triangle and divides them into proportional segments (AD/DB = AE/EC), then that line must be parallel to the third side, BC.
How does this relate to the Midsegment Theorem?
The Midsegment Theorem is a special case of the Triangle Proportionality Theorem where D and E are the midpoints of AB and AC, making AD/DB = AE/EC = 1, and the segment DE is always exactly half the length of BC.

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