The Basic Proportionality Theorem
Simple Explanation
Also known as Thales' theorem: if a line is drawn parallel to one side of a triangle, cutting the other two sides, it divides those two sides in exactly the same ratio: AD/DB = AE/EC.
Why Do We Need It?
This theorem is the key tool for proving many other results about similar triangles, and it directly explains why parallel cuts always create proportional pieces.
See It
Triangle ABC with a line segment DE parallel to BC, D on AB and E on AC
Formula
The Basic Proportionality Theorem (Thales' Theorem)
If DE ∥ BC in △ABC (D on AB, E on AC), then AD/DB = AE/EC
A line drawn parallel to one side of a triangle, cutting the other two sides, divides those two sides in exactly the same ratio.
- D, E
- — the points where the parallel line crosses sides AB and AC
- AD, DB, AE, EC
- — the four segments the parallel line creates on the two cut sides
When to use it: Whenever a line parallel to one side of a triangle cuts the other two sides, and you need to relate the resulting segment lengths.
Worked Example
Apply the basic proportionality theorem
In △ABC, DE ∥ BC, with D on AB and E on AC. If AD=4, DB=6, and AE=6, find EC.
Why Does This Work?
Stated simply: since DE ∥ BC, △ADE and △ABC share the same angle at A, and DE ∥ BC makes the angles at D and B equal (corresponding angles on parallel lines) — so △ADE ~ △ABC by AA similarity. Their sides are proportional: AD/AB = AE/AC. Since AB = AD + DB and AC = AE + EC, this proportion rearranges algebraically into exactly AD/DB = AE/EC.
Real-Life Example
Evenly spaced shelf supports
A carpenter cuts parallel shelf supports at fixed angles inside a triangular bracket frame.
The basic proportionality theorem guarantees each parallel cut divides the frame's slanted edges in the same fixed ratio, letting the carpenter predict every measurement in advance.
Practice
In △ABC, DE ∥ BC. AD=3, DB=9, AE=5. Find EC.
HardCommon mistake
Using AD/AB = AE/EC (mixing a whole side with a partial segment) instead of the correct AD/DB = AE/EC (comparing the two partial segments on each side to each other).
Quick Review
- If DE ∥ BC in △ABC, then AD/DB = AE/EC.
- Proof idea: DE ∥ BC makes △ADE ~ △ABC (AA), and algebraic rearrangement gives the segment ratio.
- Also called Thales' theorem.