Thales Theorem

Thales Theorem is a theorem that is credited to Thales (after Thales of Miletus, (c. 620 - c. 546 BC). It says that “If a straight line is drawn parallel to one of the sides of a triangle, then it cuts the other sides of the triangle, or these produced, proportionally; and, if the sides of the triangle, or the sides produced, are cut proportionally, then the line joining the points of section is parallel to the remaining side of the triangle”

Example:

Given DE||BC, we want to prove that BD/AD = CE/AE.

Join BE, CD. Then triangles BDE and CDE have equal areas by Elements I.38: Area (BDE) = Area (CDE). Triangles ADE and BDE share an altitude (from E) and, therefore, by Elements VI.1, Area (ADE)/Area (BDE) = AD/BD.

Turning to triangles CDE and ADE, we similarly observe that Area (ADE)/Area (CDE) = AE/CE. But, since Area (BDE) = Area (CDE), we also have Area (ADE)/Area (BDE) = Area (ADE)/Area (CDE), implying AD/BD = AE/CE, as required.

Now for the converse. Given AD/BD = AE/CE, prove that DE||BC.

For a proof, note that the steps above are reversible. First we have Area (ADE)/Area (BDE) = AD/BD and Area (ADE)/Area (CDE) = AC/CE, implying Area (ADE)/Area (BDE) = Area (ADE)/Area (CDE). I.e., Area (BDE) = Area (CDE), while the two triangle share the base DE. By Elements I.39, DE||BC.

Note that AD/BD = AC/CE is equivalent to AB/BD = AC/CE. In particular, triangles ABC and ADE are similar so that their bases are in the same ratio: AB/BD = AC/CE = BC/DE. As a consequence, if DE is a midline of ΔABC, i.e., if AD = BD and AE = CE, then DE is half BC: DE = BC/2.

Application

To prove Thales Theorem I made an experiment at Cideb

1- I looked for a post and I measured its shade. (257 cm)

2- I measured my height (155 cm)

3- I measured my shade (142 cm)

4- Once I knew the data, I replaced them at Thales Theorem formula

BD/AD = CE/AE.

BD: Post’s Height

AD: Post’s Shade

CE: My height

AE: My shade

5- Then, I replaced the data in the equation

BD/257cm = 155cm/142cm

Isolating BD:

BD = (155) (257)/142

That gives us the result BD=280.5cm

Answer: The post’s height is 280.5 cm

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