§6.prop.25τῷ δοθέντι εὐθυγράμμῳ ὅμοιον καὶ ἄλλῳ τῷ δοθέντι ἴσον τὸ αὐτὸ συστήσασθαι.
To construct one and the same figure similar to a given rectilinear figure and equal to another given rectilinear figure.
ἔστω τὸ μὲν δοθὲν εὐθύγραμμον, ᾧ δεῖ ὅμοιον συστήσασθαι, τὸ ΑΒΓ, ᾧ δὲ δεῖ ἴσον, τὸ Δ· δεῖ δὴ τῷ μὲν ΑΒΓ ὅμοιον, τῷ δὲ Δ ἴσον τὸ αὐτὸ συστήσασθαι.
Let the given rectilinear figure to which it is required to construct a similar one be ΑΒΓ, and that to which it is required to construct an equal one be Δ; it is required indeed to construct one and the same figure similar to ΑΒΓ and equal to Δ.
παραβεβλήσθω γὰρ παρὰ μὲν τὴν ΒΓ τῷ ΑΒΓ τριγώνῳ ἴσον παραλληλόγραμμον τὸ ΒΕ, παρὰ δὲ τὴν ΓΕ τῷ Δ ἴσον παραλληλόγραμμον τὸ ΓΜ ἐν γωνίᾳ τῇ ὑπὸ ΖΓΕ, ἥ ἐστιν ἴση τῇ ὑπὸ ΓΒΛ. ἐπʼ εὐθείας ἄρα ἐστὶν ἡ μὲν ΒΓ τῇ ΓΖ, ἡ δὲ ΛΕ τῇ ΕΜ. καὶ εἰλήφθω τῶν ΒΓ, ΓΖ μέση ἀνάλογον ἡ ΗΘ, καὶ ἀναγεγράφθω ἀπὸ τῆς ΗΘ τῷ ΑΒΓ ὅμοιόν τε καὶ ὁμοίως κείμενον τὸ ΚΗΘ.
καὶ ἐπεί ἐστιν ὡς ἡ ΒΓ πρὸς τὴν ΗΘ, οὕτως ἡ ΗΘ πρὸς τὴν ΓΖ, ἐὰν δὲ τρεῖς εὐθεῖαι ἀνάλογον ὦσιν, ἔστιν ὡς ἡ πρώτη πρὸς τὴν τρίτην, οὕτως τὸ ἀπὸ τῆς πρώτης εἶδος πρὸς τὸ ἀπὸ τῆς δευτέρας τὸ ὅμοιον καὶ ὁμοίως ἀναγραφόμενον, ἔστιν ἄρα ὡς ἡ ΒΓ πρὸς τὴν ΓΖ, οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΚΗΘ τρίγωνον.
For let there be applied to ΒΓ a parallelogram ΒΕ equal to the triangle ΑΒΓ, and to ΓΕ a parallelogram ΓΜ equal to Δ, in an angle ΖΓΕ which is equal to the angle ΓΒΛ. Therefore ΒΓ is in a straight line with ΓΖ, and ΛΕ with ΕΜ. And let a mean proportional ΗΘ be taken of ΒΓ, ΓΖ, and let there be described on ΗΘ, similar and similarly situated to ΑΒΓ, the figure ΚΗΘ. And since as ΒΓ is to ΗΘ, so is ΗΘ to ΓΖ, and if three straight lines are proportional, as the first is to the third, so is the figure described on the first to the similar and similarly described figure on the second, therefore, as ΒΓ is to ΓΖ, so is the triangle ΑΒΓ to the triangle ΚΗΘ.
ἀλλὰ καὶ ὡς ἡ ΒΓ πρὸς τὴν ΓΖ, οὕτως τὸ ΒΕ παραλληλόγραμμον πρὸς τὸ ΕΖ παραλληλόγραμμον.
But also as ΒΓ is to ΓΖ, so is the parallelogram ΒΕ to the parallelogram ΕΖ.
καὶ ὡς ἄρα τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΚΗΘ τρίγωνον, οὕτως τὸ ΒΕ παραλληλόγραμμον πρὸς τὸ ΕΖ παραλληλόγραμμον· ἐναλλὰξ ἄρα ὡς τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΒΕ παραλληλόγραμμον, οὕτως τὸ ΚΗΘ τρίγωνον πρὸς τὸ ΕΖ παραλληλόγραμμον.
Therefore also, as the triangle ΑΒΓ is to the triangle ΚΗΘ, so is the parallelogram ΒΕ to the parallelogram ΕΖ; alternate, therefore, as the triangle ΑΒΓ is to the parallelogram ΒΕ, so is the triangle ΚΗΘ to the parallelogram ΕΖ.
ἴσον δὲ τὸ ΑΒΓ τρίγωνον τῷ ΒΕ παραλληλογράμμῳ· ἴσον ἄρα καὶ τὸ ΚΗΘ τρίγωνον τῷ ΕΖ παραλληλογράμμῳ.
But the triangle ΑΒΓ is equal to the parallelogram ΒΕ; therefore the triangle ΚΗΘ is also equal to the parallelogram ΕΖ.
ἀλλὰ τὸ ΕΖ παραλληλόγραμμον τῷ Δ ἐστιν ἴσον· καὶ τὸ ΚΗΘ ἄρα τῷ Δ ἐστιν ἴσον.
But the parallelogram ΕΖ is equal to Δ; therefore ΚΗΘ is also equal to Δ.
ἔστι δὲ τὸ ΚΗΘ καὶ τῷ ΑΒΓ ὅμοιον.
And ΚΗΘ is also similar to ΑΒΓ.
τῷ ἄρα δοθέντι εὐθυγράμμῳ τῷ ΑΒΓ ὅμοιον καὶ ἄλλῳ τῷ δοθέντι τῷ Δ ἴσον τὸ αὐτὸ συνέσταται τὸ ΚΗΘ· ὅπερ ἔδει ποιῆσαι.
Therefore, one and the same figure ΚΗΘ has been constructed similar to the given rectilinear figure ΑΒΓ and equal to the other given figure Δ; which was to be done.