Humanitext Reader

Euclid · Elements §6.prop.21

Similarity of Figures Similar to the Same Figure

Passage 100 of 316 · Greek

Summary

Proves that two rectilinear figures similar to the same rectilinear figure are also similar to one another, using the transitivity of equiangularity and proportionality of corresponding sides.

§6.prop.21τὰ τῷ αὐτῷ εὐθυγράμμῳ ὅμοια καὶ ἀλλήλοις ἐστὶν ὅμοια.
Figures which are similar to the same rectilinear figure are also similar to one another.
ἔστω γὰρ ἑκάτερον τῶν Α, Β εὐθυγράμμων τῷ Γ ὅμοιον· λέγω, ὅτι καὶ τὸ Α τῷ Β ἐστιν ὅμοιον.
For let each of the rectilinear figures A, B be similar to Γ; I say that A is also similar to B.
ἐπεὶ γὰρ ὅμοιόν ἐστι τὸ Α τῷ Γ, ἰσογώνιόν τέ ἐστιν αὐτῷ καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον ἔχει.
¦5 For since A is similar to Γ, it is both equiangular with it and has the sides about the equal angles proportional.
πάλιν, ἐπεὶ ὅμοιόν ἐστι τὸ Β τῷ Γ, ἰσογώνιόν τέ ἐστιν αὐτῷ καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον ἔχει.
Again, since B is similar to Γ, it is both equiangular with it and has the sides about the equal angles proportional.
ἑκάτερον ἄρα τῶν Α, Β τῷ Γ ἰσογώνιόν τέ ἐστι καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον ἔχει.
Therefore each ¦10 of A, B is both equiangular with Γ and has the sides about the equal angles proportional.
ὅμοιον ἄρα ἐστὶ τὸ Α τῷ Β·
Therefore A is ¦15 similar to B.
ὅπερ ἔδει δεῖξαι.
Which was to be demonstrated.

Notes

  1. 6.prop.21τὰ τῷ αὐτῷ εὐθυγράμμῳ ὅμοια — The neuter plural adjective with article τὰ ... ὅμοια is used substantively to mean 'similar figures' and functions as the subject. τῷ αὐτῷ εὐθυγράμμῳ is in the dative case, depending on the adjective ὅμοια, and expresses the standard of comparison ('to the same rectilinear figure').
  2. 6.prop.21τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον ἔχει — The direct object of ἔχει (has) is τὰς ... πλευράς (the sides), and ἀνάλογον (proportionally) functions adverbially or as an object complement. Thus, the phrase literally means 'has the sides about the equal angles proportional'.

Cite this passage

Euclid, Elements §6.prop.21. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:6.prop.21

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