Thales' theorem – parallel lines in a triangle
MathematicsPlane GeometryAges 13–14
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Plane Geometry
Triangle ABC with a line DE crossing sides AB and AC. Drag D along AB (E follows so that DE ∥ BC) to see the ratios AD/DB = AE/EC and AD/AB = AE/AC stay equal. Switch to free mode to drag D and E independently and test the converse: when the two ratios are equal, DE ∥ BC. All three vertices of the triangle can be dragged too.
Lesson: Thales' theorem in a triangle, its converse, and its corollary
What it shows
A line DE cuts sides AB and AC of triangle ABC. Thales' theorem, also called the triangle proportionality theorem, states that if DE is parallel to BC, it divides the two sides proportionally: AD/DB = AE/EC and AD/AB = AE/AC. The converse says that equal ratios force DE to be parallel to BC. The corollary compares the parallel segment with the base, DE/BC = AD/AB, a ratio that is often confused with AD/DB. Lengths are measured on a centimeter grid, so each ratio can be checked from the actual measurements.
How to use
Start in DE ∥ BC mode and move D along AB with Position of D on AB, reshaping the triangle by dragging A, B and C; the ratio pair always stays equal. Switch between AD/DB, AD/AB and DB/AB, and tick Show corollary DE/BC. Then choose Free drag and move D and E independently: students try to make the two ratios equal and observe that DE becomes parallel to BC exactly then.
Parameters you can change
- Position of D on AB (AD/AB) 0.1–0.9
- Ratio form shown AD/DB = AE/EC, AD/AB = AE/AC, DB/AB = EC/AC
- Mode DE always parallel to BC, Drag D, E freely (converse)
- Show corollary DE/BC
Questions to explore
- If AD/DB = 2 and DE is parallel to BC, what is AE/EC?
- Why is DE/BC equal to AD/AB rather than AD/DB?
- Given only the lengths AD, DB, AE and EC, how can you decide whether DE is parallel to BC?
