The Intercept Theorem (Thales) — Proof

What just happened?

Two parallel lines crossed both arms of the angle and marked off segments on them. Although the segments on one arm are longer than on the other, the ratios of corresponding segments are identical — and they stay that way wherever you move the lines.

The intercept theorem:

If we cut the arms of an angle with two parallel lines, then the ratio of the segments marked off on one arm equals the ratio of the corresponding segments on the other arm:

The theorem is attributed to Thales of Miletus (c. 624–546 BC), a Greek philosopher and mathematician, regarded as the first mathematician in history.

The invariant — the proportion of segments:

The value of this ratio depends only on the position of the lines relative to the vertex, not on which arm we choose. That is why the segments are proportional, not equal: they would be equal only if the arms were symmetric with respect to the lines.

Why does it work?

The parallel lines form, with the arms of the angle, similar triangles (the same angles). In similar figures, corresponding sides are proportional — that is where the equality of ratios comes from. The intercept theorem and the similarity of triangles are two sides of the same idea.

The legend of the pyramid:

Legend has it that Thales measured the height of an Egyptian pyramid from the length of its shadow — comparing it with the shadow of a stick of known height. The same reasoning about proportions: $\tfrac{\text{height}}{\text{shadow}}$ is the same for the pyramid and for the stick.

Why you need this for the exam:

The intercept theorem and its converse are the basis of many constructions (e.g. dividing a segment into equal parts) and proofs in geometry — essential exam material. Trick: write down the corresponding segments and set their ratios equal.

Remember: parallel lines crossing the arms of an angle divide them into proportional segments — $\tfrac{|OA|}{|OB|}=\tfrac{|OA'|}{|OB'|}$, regardless of the position of the lines.