A line parallel to one side of a triangle divides the other two sides proportionally.
Key formulas
Parallel trigger
If DE ∥ BC with D on AB and E on AC, then AD/DB = AE/EC = DE/BC.
Example
AD=3, DB=6, AE=2 → EC=4 because 3/6=2/EC.
Converse for parallel
Equal ratios imply DE ∥ BC — useful when you need to prove parallelism.
Worked example
Worked example
Problem. In △ABC, DE ∥ BC with D on AB and E on AC. AD = 3, DB = 6, AE = 2. Find EC.
Solution. AD/DB = AE/EC → 3/6 = 2/EC → EC = 4.
Whole sides
Also AD/AB = AE/AC.
Review quiz
DE∥BC, AD=4, AB=10, AE=6. Then AC is:
Practice questions
- AD=4, AB=10, AE=6. Find AC if DE∥BC.
- Why does parallelism create equal ratios?
Study notes
Is this the same as similar triangles?
It follows from AA similarity of △ADE and △ABC.
More in Plane Geometry
- Circle tangents: the radius is perpendicular
- Right-triangle lengths: Pythagoras and altitude
- Similar triangles: matching sides in proportion
- Chord theorem: perpendicular from the centre
- Isosceles triangles and equal base angles
- Alternate segment theorem with a short proof
- Converse of Pythagoras: proving a right angle
- Heron’s formula when height is hidden