§3.prop.33ἐπὶ τῆς δοθείσης εὐθείας γράψαι τμῆμα κύκλου δεχόμενον γωνίαν ἴσην τῇ δοθείσῃ γωνίᾳ εὐθυγράμμῳ.
On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilinear angle.
ἔστω ἡ δοθεῖσα εὐθεῖα ἡ ΑΒ, ἡ δὲ δοθεῖσα γωνία εὐθύγραμμος ἡ πρὸς τῷ Γ· δεῖ δὴ ἐπὶ τῆς δοθείσης εὐθείας τῆς ΑΒ γράψαι τμῆμα κύκλου δεχόμενον γωνίαν ἴσην τῇ πρὸς τῷ Γ.
ἡ δὴ πρὸς τῷ Γ ἤτοι ὀξεῖά ἐστιν ἢ ὀρθὴ ἢ ἀμβλεῖα·
Let ΑΒ be the given straight line, and the angle at Γ the given rectilinear angle; thus it is required on the given straight line ΑΒ to describe a segment of a circle admitting an angle equal to the angle at Γ. Now the angle at Γ is either acute, right, or obtuse.
ἔστω πρότερον ὀξεῖα, καὶ ὡς ἐπὶ τῆς πρώτης καταγραφῆς συνεστάτω πρὸς τῇ ΑΒ εὐθείᾳ καὶ τῷ α σημείῳ τῇ πρὸς τῷ Γ γωνίᾳ ἴση ἡ ὑπὸ ΒΑΔ·
Let it first be acute, and, as in the first figure, let there be constructed on the straight line ΑΒ and at the point Α, the angle ΒΑΔ equal to the angle at Γ; therefore the angle ΒΑΔ is also acute.
ὀξεῖα ἄρα ἐστὶ καὶ ἡ ὑπὸ ΒΑΔ. ἤχθω τῇ ΔΑ πρὸς ὀρθὰς ἡ ΑΕ, καὶ τετμήσθω ἡ ΑΒ δίχα κατὰ τὸ Ζ, καὶ ἤχθω ἀπὸ τοῦ Ζ σημείου τῇ ΑΒ πρὸς ὀρθὰς ἡ ΖΗ, καὶ ἐπεζεύχθω ἡ ΗΒ.
καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΖ τῇ ΖΒ, κοινὴ δὲ ἡ ΖΗ, δύο δὴ αἱ ΑΖ, ΖΗ δύο ταῖς ΒΖ, ΖΗ ἴσαι εἰσίν· καὶ γωνία ἡ ὑπὸ ΑΖΗ τῇ ὑπὸ ΒΖΗ ἴση· βάσις ἄρα ἡ ΑΗ βάσει τῇ ΒΗ ἴση ἐστίν.
Let ΑΕ be drawn at right angles to ΔΑ, let ΑΒ be bisected at Ζ, let ΖΗ be drawn from the point Ζ at right angles to ΑΒ, and let ΗΒ be joined. And since ΑΖ is equal to ΖΒ, and ΖΗ is common, the two straight lines ΑΖ, ΖΗ are equal to the two straight lines ΒΖ, ΖΗ; and the angle ΑΖΗ is equal to the angle ΒΖΗ; therefore the base ΑΗ is equal to the base ΒΗ.
ὁ ἄρα κέντρῳ μὲν τῷ Η διαστήματι δὲ τῷ ΗΑ κύκλος γραφόμενος ἥξει καὶ διὰ τοῦ Β. γεγράφθω καὶ ἔστω ὁ ΑΒΕ, καὶ ἐπεζεύχθω ἡ ΕΒ. ἐπεὶ οὖν ἀπʼ ἄκρας τῆς ΑΕ διαμέτρου ἀπὸ τοῦ Α τῇ ΑΕ πρὸς ὀρθάς ἐστιν ἡ ΑΔ, ἡ ΑΔ ἄρα ἐφάπτεται τοῦ ΑΒΕ κύκλου·
Therefore the circle described with center Η and distance ΗΑ will pass also through Β. Let it be described, and let it be ΑΒΕ, and let ΕΒ be joined. Since then, from the end Α of the diameter ΑΕ, ΑΔ is at right angles to ΑΕ, therefore ΑΔ touches the circle ΑΒΕ.
ἐπεὶ οὖν κύκλου τοῦ ΑΒΕ ἐφάπτεταί τις εὐθεῖα ἡ ΑΔ, καὶ ἀπὸ τῆς κατὰ τὸ Α ἁφῆς εἰς τὸν ΑΒΕ κύκλον διῆκταί τις εὐθεῖα ἡ ΑΒ, ἡ ἄρα ὑπὸ ΔΑΒ γωνία ἴση ἐστὶ τῇ ἐν τῷ ἐναλλὰξ τοῦ κύκλου τμήματι γωνίᾳ τῇ ὑπὸ ΑΕΒ. ἀλλʼ ἡ ὑπὸ ΔΑΒ τῇ πρὸς τῷ Γ ἐστιν ἴση· καὶ ἡ πρὸς τῷ Γ ἄρα γωνία ἴση ἐστὶ τῇ ὑπὸ ΑΕΒ.
ἐπὶ τῆς δοθείσης ἄρα εὐθείας τῆς ΑΒ τμῆμα κύκλου γέγραπται τὸ ΑΕΒ δεχόμενον γωνίαν τὴν ὑπὸ ΑΕΒ ἴσην τῇ δοθείσῃ τῇ πρὸς τῷ Γ.
ἀλλὰ δὴ ὀρθὴ ἔστω ἡ πρὸς τῷ Γ·
Since then a straight line ΑΔ touches the circle ΑΒΕ, and from the point of contact Α there has been drawn across the circle ΑΒΕ a straight line ΑΒ, therefore the angle ΔΑΒ is equal to the angle ΑΕΒ in the alternate segment of the circle. But the angle ΔΑΒ is equal to the angle at Γ; therefore the angle at Γ is also equal to the angle ΑΕΒ. Therefore, on the given straight line ΑΒ, a segment of a circle ΑΕΒ has been described admitting an angle ΑΕΒ equal to the given angle at Γ.
καὶ δέον πάλιν ἔστω ἐπὶ τῆς ΑΒ γράψαι τμῆμα κύκλου δεχόμενον γωνίαν ἴσην τῇ πρὸς τῷ Γ ὀρθῇ.
Next, let the angle at Γ be a right angle; and let it again be required on ΑΒ to describe a segment of a circle admitting an angle equal to the right angle at Γ.
συνεστάτω τῇ πρὸς τῷ Γ ὀρθῇ γωνίᾳ ἴση ἡ ὑπὸ ΒΑΔ, ὡς ἔχει ἐπὶ τῆς δευτέρας καταγραφῆς, καὶ τετμήσθω ἡ ΑΒ δίχα κατὰ τὸ Ζ, καὶ κέντρῳ τῷ Ζ, διαστήματι δὲ ὁποτέρῳ τῶν ΖΑ, ΖΒ, κύκλος γεγράφθω ὁ ΑΕΒ.
ἐφάπτεται ἄρα ἡ ΑΔ εὐθεῖα τοῦ ΑΒΕ κύκλου διὰ τὸ ὀρθὴν εἶναι τὴν πρὸς τῷ Α γωνίαν.
Let the angle ΒΑΔ be constructed equal to the right angle at Γ, as in the second figure, and let ΑΒ be bisected at Ζ, and with center Ζ and distance either of the two, ΖΑ, ΖΒ, let a circle ΑΕΒ be described. Therefore the straight line ΑΔ touches the circle ΑΒΕ, because the angle at Α is a right angle.
καὶ ἴση ἐστὶν ἡ ὑπὸ ΒΑΔ γωνία τῇ ἐν τῷ ΑΕΒ τμήματι· ὀρθὴ γὰρ καὶ αὐτὴ ἐν ἡμικυκλίῳ οὖσα.
And the angle ΒΑΔ is equal to the angle in the segment ΑΕΒ; for it too is a right angle, being in a semicircle.
ἀλλὰ καὶ ἡ ὑπὸ ΒΑΔ τῇ πρὸς τῷ Γ ἴση ἐστίν. καὶ ἡ ἐν τῷ ΑΕΒ ἄρα ἴση ἐστὶ τῇ πρὸς τῷ Γ.
γέγραπται ἄρα πάλιν ἐπὶ τῆς ΑΒ τμῆμα κύκλου τὸ ΑΕΒ δεχόμενον γωνίαν ἴσην τῇ πρὸς τῷ Γ.
ἀλλὰ δὴ ἡ πρὸς τῷ Γ ἀμβλεῖα ἔστω·
But the angle ΒΑΔ is also equal to the angle at Γ; therefore the angle in ΑΕΒ is also equal to the angle at Γ. Therefore, again, a segment of a circle ΑΕΒ has been described on ΑΒ admitting an angle equal to the angle at Γ.
καὶ συνεστάτω αὐτῇ ἴση πρὸς τῇ ΑΒ εὐθείᾳ καὶ τῷ Α σημείῳ ἡ ὑπὸ ΒΑΔ, ὡς ἔχει ἐπὶ τῆς τρίτης καταγραφῆς, καὶ τῇ ΑΔ πρὸς ὀρθὰς ἤχθω ἡ ΑΕ, καὶ τετμήσθω πάλιν ἡ ΑΒ δίχα κατὰ τὸ Ζ, καὶ τῇ ΑΒ πρὸς ὀρθὰς ἤχθω ἡ ΖΗ, καὶ ἐπεζεύχθω ἡ ΗΒ.
καὶ ἐπεὶ πάλιν ἴση ἐστὶν ἡ ΑΖ τῇ ΖΒ, καὶ κοινὴ ἡ ΖΗ, δύο δὴ αἱ ΑΖ, ΖΗ δύο ταῖς ΒΖ, ΖΗ ἴσαι εἰσίν· καὶ γωνία ἡ ὑπὸ ΑΖΗ γωνίᾳ τῇ ὑπὸ ΒΖΗ ἴση· βάσις ἄρα ἡ ΑΗ βάσει τῇ ΒΗ ἴση ἐστίν·
Next, let the angle at Γ be an obtuse angle; and let there be constructed equal to it, on the straight line ΑΒ and at the point Α, the angle ΒΑΔ, as in the third figure, and let ΑΕ be drawn at right angles to ΑΔ, and let ΑΒ again be bisected at Ζ, and let ΖΗ be drawn at right angles to ΑΒ, and let ΗΒ be joined. And since, again, ΑΖ is equal to ΖΒ, and ΖΗ is common, the two straight lines ΑΖ, ΖΗ are equal to the two straight lines ΒΖ, ΖΗ; and the angle ΑΖΗ is equal to the angle ΒΖΗ; therefore the base ΑΗ is equal to the base ΒΗ; therefore the circle described with center Η and distance ΗΑ will pass also through Β.
ὁ ἄρα κέντρῳ μὲν τῷ η διαστήματι δὲ τῷ ΗΑ κύκλος γραφόμενος ἥξει καὶ διὰ τοῦ Β. ἐρχέσθω ὡς ὁ ΑΕΒ. καὶ ἐπεὶ τῇ ΑΕ διαμέτρῳ ἀπʼ ἄκρας πρὸς ὀρθάς ἐστιν ἡ ΑΔ, ἡ ΑΔ ἄρα ἐφάπτεται τοῦ ΑΕΒ κύκλου.
Let it go as ΑΕΒ. And since ΑΔ is at right angles to the diameter ΑΕ from its end, therefore ΑΔ touches the circle ΑΕΒ.
καὶ ἀπὸ τῆς κατὰ τὸ Α ἐπαφῆς διῆκται ἡ ΑΒ· ἡ ἄρα ὑπὸ ΒΑΔ γωνία ἴση ἐστὶ τῇ ἐν τῷ ἐναλλὰξ τοῦ κύκλου τμήματι τῷ ΑΘΒ συνισταμένῃ γωνίᾳ.
And from the contact at Α, ΑΒ has been drawn; therefore the angle ΒΑΔ is equal to the angle constructed in the alternate segment of the circle, ΑΘΒ.
ἀλλʼ ἡ ὑπὸ ΒΑΔ γωνία τῇ πρὸς τῷ Γ ἴση ἐστίν. καὶ ἡ ἐν τῷ ΑΘΒ ἄρα τμήματι γωνία ἴση ἐστὶ τῇ πρὸς τῷ Γ.
ἐπὶ τῆς ἄρα δοθείσης εὐθείας τῆς ΑΒ γέγραπται τμῆμα κύκλου τὸ ΑΘΒ δεχόμενον γωνίαν ἴσην τῇ πρὸς τῷ Γ· ὅπερ ἔδει ποιῆσαι.
But the angle ΒΑΔ is equal to the angle at Γ; therefore the angle in the segment ΑΘΒ is also equal to the angle at Γ. Therefore, on the given straight line ΑΒ, a segment of a circle ΑΘΒ has been described admitting an angle equal to the angle at Γ; which was to be done.