Commentary on Proposition VI.20

Back to Proposition VI.20.

An alternative proof of the second part of this proposition given after the Porisms is relegated by August and Heiberg to an Appendix as an interpolation. It is shorter than the proof in the text, and is the only one given by many editors, including Clavius, Billingsley, Barrow and Simson. It runs as follows:

“We will now also prove that the triangles are homologous in another and an easier manner.

A B C D E F G H K L

Again, let the polygons ABCDE, FGHKL be set out, and let BE, EC, GL, LH be joined.

I say that, as the triangle ABE is to FGL, so is EBC to LGH and CDE to HKL.

For, since the triangle ABE is similar to the triangle FGL, the triangle ABE has to the triangle FGL a ratio duplicate of that which BE has to GL.
For the same reason also
the triangle BEC has to the triangle GLH a ratio duplicate of that which BE has to GL.
Therefore, as the triangle ABE is to the triangle FGL, so is BEC to GLH.
Again, since the triangle EBC is similar to the triangle LGH,
EBC has to LGH a ratio duplicate of that which the straight line CE has to HL.
For the same reason also
the triangle ECD has to the triangle LHK a ratio duplicate of that which CE has to HL.
Therefore, as the triangle EBC is to LGH, so is ECD to LHK.
But it was proved that, as EBC is to LGH, so also is ABE to FGL.
Therefore also, as ABE is to FGL, so is BEC to GLH and ECD to LHK.
Q. E. D."

Now Euclid cannot fail to have noticed that the second part of his proposition could be proved in this way. It seems therefore that, in giving the other and longer method, he deliberately wished to avoid using the result of VI. 19, preferring to prove the first two parts of the theorem, as they can be proved, independently of any relation between the areas of similar triangles.

The first part of the Porism, stating that the theorem is true of quadrilaterals, would be superfluous but for the fact that technically, according to Book I. Def. 19, the term “polygon” (or figure of many sides, πολύπλευρον) used in the enunciation of the proposition is confined to rectilineal figures of more than four sides, so that a quadrilateral might seem to be excluded. The mention of the triangle in addition fills up the tale of “similar rectilineal figures.”

The second Porism, Theon’s interpolation, given in the text by the editors, but bracketed by Heiberg, is as follows:

“And, if we take $O$ a third proportional to AB, FG, then BA has to $O$ a ratio duplicate of that which AB has to FG. But the polygon has also to the polygon, or the quadrilateral to the quadrilateral, a ratio duplicate of that which the corresponding side has to the corresponding side, that is AB to FG; and this was proved in the case of triangles also; so that it is also manifest generally that, if three straight lines be proportional, as the first is to the third, so will the figure described on the first be to the similar and similarly described figure on the second.”