Traditional traps and their applications

Pitfall 1: Confusing the theorem and its converse

Using the direct theorem when no parallelism is known, or using the converse when the parallelism is already given in the question: in the latter case, Thales’ theorem can be applied directly; the converse is unnecessary.

Pitfall 2: Mixing up the ratios

Writing AM/AC or AN/AB instead of AM/AB and AN/AC. To avoid this error, make sure you note which length lies on which line before setting up the proportion.

Pitfall 3: Forgetting to check the alignment

Thales’ theorem only applies if the points are correctly aligned with the common vertex (A, M, B aligned; A, N, C aligned). Without this alignment, the configuration does not satisfy the conditions for Thales’ theorem, even if the lines appear parallel in the diagram.

Pitfall 4: rounding off too early

When a calculation produces a number with many decimal places, it is best to retain the fractions or several decimal places right up to the final result, so as not to distort the answer.

Practical application: measuring an inaccessible height

To estimate the height of a tree, one plants a vertical stick of known height near the tree, measure the shadows cast by the sun (that of the stick and that of the tree), then use Thales’s proportionality between the two triangles formed by the sun, the stick (or the tree) and its shadow.

Table: when to use what?

Situation	Tool to use
Parallelism is given; we are looking for a length	Thales’ theorem (direct)
The lengths are known; we want to prove parallelism	Reciprocal of Thales’ theorem