The Basic Proportionality Theorem

Draw a triangle ABC. Draw a line DE parallel to BC, cutting AB at D and AC at E. Measure, and you find the two sides are cut in the same ratio:

AD/DB = AE/EC

This is the Basic Proportionality Theorem (BPT), also called Thales' theorem after the Greek mathematician Thales. In short: a line parallel to one side cuts the other two sides in the same ratio.

Why it is true: the book's proof uses areas.

• Triangles ADE and BDE have the same height from E, so their areas are in the ratio of their bases: ar(ADE)/ar(BDE) = AD/DB.
• In the same way, ar(ADE)/ar(DEC) = AE/EC.
• Triangles BDE and DEC sit on the same base DE between the same parallels DE and BC, so their areas are equal.
• So the two ratios are equal: AD/DB = AE/EC.

(Don't 'prove' BPT using similar triangles. The book uses BPT to prove the similarity rules, so that would be going round in a circle.)

Other forms you can use: turn both fractions upside down (DB/AD = EC/AE), or compare a piece with the whole side: AD/AB = AE/AC.

The converse (Theorem 6.2): if a line cuts two sides of a triangle in the same ratio, then it is parallel to the third side. So:

• parallel ⇒ equal ratios (BPT)
• equal ratios ⇒ parallel (converse)

Finding a length: DE ∥ BC, BD = 7.2, AE = 1.8 and EC = 5.4. Then AD/7.2 = 1.8/5.4 = 1/3, so AD = 2.4 cm.