The converse of Thales’ theorem

What is the reciprocal used for?

Thales’ theorem allows us to calculate a length when we already know that two lines are parallel. The reciprocal does the opposite: it allows us to prove that two lines are parallel, based on known lengths.

The statement

Let A be a point, and let M and B lie on the same line as A (M lies on [AB]), and N and C lie on the same line as A (N lies on [AC]). If:

AM/AB = AN/AC

then the lines (MN) and (BC) are parallel.

Example

Given that AB = 10 cm, AM = 4 cm, AC = 15 cm and AN = 6 cm.

We calculate the two ratios: AM/AB = 4/10 = 0.4 AN/AC = 6/15 = 0.4

The two ratios are equal, so by Thales’s reciprocal theorem, (MN) // (BC).

Table of conditions to be met

Condition	Explanation
Alignment	A, M, B in a straight line AND A, N, C in a straight line
Equality of ratios	AM/AB = AN/AC (calculated separately)
Conclusion	(MN) is parallel to (BC)

Pitfall: do not confuse with the direct theorem

Thales’ theorem (direct) starts from a known parallelism to calculate a length. The converse starts from known lengths to prove a parallelism. These are not the same assumptions nor the same conclusions: reading the question carefully will help you determine which one to use.