Identifying a Thales configuration

Thales’ theorem – what is it used for?

Thales’ theorem allows us to calculate lengths in figures or where parallel lines intersect other lines. It is used extensively in geometry, but also to estimate a height (of a tree or a building) or a distance that cannot be measured directly.

The basic configuration

We start with a point A, which is the common vertex of two lines. On one of these, we mark point B and then point M (M lies on the segment [AB]). On the other line, we place point C and then point N (N lies on the segment [AC]). If the line (MN) is parallel to the line (BC), we say that we have a Thales’ configuration.

Two possible forms

Configuration	Description
Triangle	M and N lie between A and the vertices B and C; a small triangle AMN can be seen inside the larger triangle ABC
Butterfly	The lines (AB) and (AC) intersect at A, but M and N lie on opposite sides of A with respect to B and C

Example

In a triangle ABC, M is a point on [AB] and N is a point on [AC]. If (MN) // (BC), then ABC and AMN form a Thales’ configuration: the lengths AM, AB, AN, AC, MN and BC can be related by the same proportion (see the next lesson).

Things to check before applying Thales’ theorem

• A common vertex (here, A)
• The points are correctly aligned (A, M, B on one side; A, N, C on the other)
• The lines (MN) and (BC) are strictly parallel

Common pitfall: believing that it is sufficient for two lines to intersect in order to use Thales’ theorem. It is absolutely essential that line (MN) is parallel to (BC), in addition to the points being aligned; otherwise, the configuration is not valid.