§6.prop.24παντὸς παραλληλογράμμου τὰ περὶ τὴν διάμετρον παραλληλόγραμμα ὅμοιά ἐστι τῷ τε ὅλῳ καὶ ἀλλήλοις.
In any parallelogram, the parallelograms about the diameter are similar to the whole and to one another.
ἔστω παραλληλόγραμμον τὸ ΑΒΓΔ, διάμετρος δὲ αὐτοῦ ἡ ΑΓ, περὶ δὲ τὴν ΑΓ παραλληλόγραμμα ἔστω τὰ ΕΗ, ΘΚ· λέγω, ὅτι ἑκάτερον τῶν ΕΗ, ΘΚ παραλληλογράμμων ὅμοιόν ἐστι ὅλῳ τῷ ΑΒΓΔ καὶ ἀλλήλοις.
Let there be a parallelogram ΑΒΓΔ, and its diameter ΑΓ, and let the parallelograms about ΑΓ be ΕΗ, ΘΚ; I say that each of the parallelograms ΕΗ, ΘΚ is similar to the whole ΑΒΓΔ and to one another.
ἐπεὶ γὰρ τριγώνου τοῦ ΑΒΓ παρὰ μίαν τῶν πλευρῶν τὴν ΒΓ ἦκται ἡ ΕΖ, ἀνάλογόν ἐστιν ὡς ἡ ΒΕ πρὸς τὴν ΕΑ, οὕτως, ἡ ΓΖ πρὸς τὴν ΖΑ. πάλιν, ἐπεὶ τριγώνου τοῦ ΑΓΔ παρὰ μίαν τὴν ΓΔ ἦκται ἡ ΖΗ, ἀνάλογόν ἐστιν ὡς ἡ ΓΖ πρὸς τὴν ΖΑ, οὕτως ἡ ΔΗ πρὸς τὴν ΗΑ. ἀλλʼ ὡς ἡ ΓΖ πρὸς τὴν ΖΑ, οὕτως ἐδείχθη καὶ ἡ ΒΕ πρὸς τὴν ΕΑ· καὶ ὡς ἄρα ἡ ΒΕ πρὸς τὴν ΕΑ, οὕτως ἡ ΔΗ πρὸς τὴν ΗΑ, καὶ συνθέντι ἄρα ὡς ἡ ΒΑ πρὸς ΑΕ, οὕτως ἡ ΔΑ πρὸς ΑΗ, καὶ ἐναλλὰξ ὡς ἡ ΒΑ πρὸς τὴν ΑΔ, οὕτως ἡ ΕΑ πρὸς τὴν ΑΗ. τῶν ἄρα ΑΒΓΔ, ΕΗ παραλληλογράμμων ἀνάλογόν εἰσιν αἱ πλευραὶ αἱ περὶ τὴν κοινὴν γωνίαν τὴν ὑπὸ ΒΑΔ. καὶ ἐπεὶ παράλληλός ἐστιν ἡ ΗΖ τῇ ΔΓ, ἴση ἐστὶν ἡ μὲν ὑπὸ ΑΖΗ γωνία τῇ ὑπὸ ΔΓΑ· καὶ κοινὴ τῶν δύο τριγώνων τῶν ΑΔΓ, ΑΗΖ ἡ ὑπὸ ΔΑΓ γωνία· ἰσογώνιον ἄρα ἐστὶ τὸ ΑΔΓ τρίγωνον τῷ ΑΗΖ τριγώνῳ.
For since, parallel to one of the sides ΒΓ of the triangle ΑΒΓ, ΕΖ has been drawn, proportionally, as ΒΕ is to ΕΑ, so is ΓΖ to ΖΑ. Again, since, parallel to one of the sides ΓΔ of the triangle ΑΓΔ, ΖΗ has been drawn, proportionally, as ΓΖ is to ΖΑ, so is ΔΗ to ΗΑ. But, as ΓΖ is to ΖΑ, so was ΒΕ to ΕΑ also demonstrated; and therefore, as ΒΕ is to ΕΑ, so is ΔΗ to ΗΑ, and therefore, by addition, as ΒΑ is to ΑΕ, so is ΔΑ to ΑΗ, and, alternate, as ΒΑ is to ΑΔ, so is ΕΑ to ΑΗ. Therefore, of the parallelograms ΑΒΓΔ, ΕΗ, the sides about the common angle ΒΑΔ are proportional. And since ΗΖ is parallel to ΔΓ, the angle ΑΖΗ is equal to the angle ΔΓΑ; and the angle ΔΑΓ is common to the two triangles ΑΔΓ, ΑΗΖ; therefore the triangle ΑΔΓ is equiangular with the triangle ΑΗΖ.
διὰ τὰ αὐτὰ δὴ καὶ τὸ ΑΓΒ τρίγωνον ἰσογώνιόν ἐστι τῷ ΑΖΕ τριγώνῳ, καὶ ὅλον τὸ ΑΒΓΔ παραλληλόγραμμον τῷ ΕΗ παραλληλογράμμῳ ἰσογώνιόν ἐστιν.
For the same reasons indeed, the triangle ΑΓΒ is also equiangular with the triangle ΑΖΕ, and the whole parallelogram ΑΒΓΔ is equiangular with the parallelogram ΕΗ.
ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΑΔ πρὸς τὴν ΔΓ, οὕτως ἡ ΑΗ πρὸς τὴν ΗΖ, ὡς δὲ ἡ ΔΓ πρὸς τὴν ΓΑ, οὕτως ἡ ΗΖ πρὸς τὴν ΖΑ, ὡς δὲ ἡ ΑΓ πρὸς τὴν ΓΒ, οὕτως ἡ ΑΖ πρὸς τὴν ΖΕ, καὶ ἔτι ὡς ἡ ΓΒ πρὸς τὴν ΒΑ, οὕτως ἡ ΖΕ πρὸς τὴν ΕΑ. καὶ ἐπεὶ ἐδείχθη ὡς μὲν ἡ ΔΓ πρὸς τὴν ΓΑ, οὕτως ἡ ΗΖ πρὸς τὴν ΖΑ, ὡς δὲ ἡ ΑΓ πρὸς τὴν ΓΒ, οὕτως ἡ ΑΖ πρὸς τὴν ΖΕ, διʼ ἴσου ἄρα ἐστὶν ὡς ἡ ΔΓ πρὸς τὴν ΓΒ, οὕτως ἡ ΗΖ πρὸς τὴν ΖΕ. τῶν ἄρα ΑΒΓΔ, ΕΗ παραλληλογράμμων ἀνάλογόν εἰσιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας· ὅμοιον ἄρα ἐστὶ τὸ ΑΒΓΔ παραλληλόγραμμον τῷ ΕΗ παραλληλογράμμῳ.
Therefore, proportionally, as ΑΔ is to ΔΓ, so is ΑΗ to ΗΖ, and as ΔΓ is to ΓΑ, so is ΗΖ to ΖΑ, and as ΑΓ is to ΓΒ, so is ΑΖ to ΖΕ, and further, as ΓΒ is to ΒΑ, so is ΖΕ to ΕΑ. And since it was demonstrated that, as ΔΓ is to ΓΑ, so is ΗΖ to ΖΑ, and as ΑΓ is to ΓΒ, so is ΑΖ to ΖΕ, therefore, ex aequali, as ΔΓ is to ΓΒ, so is ΗΖ to ΖΕ. Therefore, of the parallelograms ΑΒΓΔ, ΕΗ, the sides about the equal angles are proportional; therefore the parallelogram ΑΒΓΔ is similar to the parallelogram ΕΗ.
διὰ τὰ αὐτὰ δὴ τὸ ΑΒΓΔ παραλληλόγραμμον καὶ τῷ ΚΘ παραλληλογράμμῳ ὅμοιόν ἐστιν· ἑκάτερον ἄρα τῶν ΕΗ, ΘΚ παραλληλογράμμων τῷ ΑΒΓΔ ὅμοιόν ἐστιν.
For the same reasons indeed, the parallelogram ΑΒΓΔ is also similar to the parallelogram ΚΘ (or ΘΚ); therefore each of the parallelograms ΕΗ, ΘΚ is similar to ΑΒΓΔ.
τὰ δὲ τῷ αὐτῷ εὐθυγράμμῳ ὅμοια καὶ ἀλλήλοις ἐστὶν ὅμοια· καὶ τὸ ΕΗ ἄρα παραλληλόγραμμον τῷ ΘΚ παραλληλογράμμῳ ὅμοιόν ἐστιν.
But figures similar to the same rectilinear figure are also similar to one another; therefore the parallelogram ΕΗ is also similar to the parallelogram ΘΚ.
παντὸς ἄρα παραλληλογράμμου τὰ περὶ τὴν διάμετρον παραλληλόγραμμα ὅμοιά ἐστι τῷ τε ὅλῳ καὶ ἀλλήλοις· ὅπερ ἔδει δεῖξαι.
Therefore, in any parallelogram, the parallelograms about the diameter are similar to the whole and to one another; which was to be demonstrated.