Page 89 - Euclid's Elements of Geometry

ST EW gþ.

ihþ the given circle BCD from the given point A. (Which is)

ELEMENTS BOOK 3

the very thing it was required to do.

Proposition 18
.
᾿Εὰν κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τοῦ κέντρου If some straight-line touches a circle, and some
ἐπὶ τὴν ἁφὴν ἐπιζευχθῇ τις εὐθεῖα, ἡ ἐπιζευχθεῖσα κάθετος (other) straight-line is joined from the center (of the cir-
ἔσται ἐπὶ τὴν ἐφαπτομένην. cle) to the point of contact, then the (straight-line) so
joined will be perpendicular to the tangent.

Α A
∆ D
Β B
Ζ Η F G

Γ C

Ε E
Κύκλου γὰρ τοῦ ΑΒΓ ἐφαπτέσθω τις εὐθεῖα ἡ ΔΕ κατὰ For let some straight-line DE touch the circle ABC at
τὸ Γ σημεῖον, καὶ εἰλήφθω τὸ κέντρον τοῦ ΑΒΓ κύκλου τὸ point C, and let the center F of circle ABC have been
Ζ, καὶ ἀπὸ τοῦ Ζ ἐπὶ τὸ Γ ἐπεζεύχθω ἡ ΖΓ· λέγω, ὅτι ἡ ΖΓ found [Prop. 3.1], and let FC have been joined from F
κάθετός ἐστιν ἐπὶ τὴν ΔΕ. to C. I say that FC is perpendicular to DE.
Εἰ γὰρ μή, ἤχθω ἀπὸ τοῦ Ζ ἐπὶ τὴν ΔΕ κάθετος ἡ ΖΗ. For if not, let FG have been drawn from F, perpen-
᾿Επεὶ οὖν ἡ ὑπὸ ΖΗΓ γωνία ὀρθή ἐστιν, ὀξεῖα ἄρα ἐστὶν dicular to DE [Prop. 1.12].
ἡ ὑπὸ ΖΓΗ· ὑπὸ δὲ τὴν μείζονα γωνίαν ἡ μείζων πλευρὰ Therefore, since angle FGC is a right-angle, (angle)
ὑποτείνει· μείζων ἄρα ἡ ΖΓ τῆς ΖΗ· ἴση δὲ ἡ ΖΓ τῇ ΖΒ· FCG is thus acute [Prop. 1.17]. And the greater angle is
μείζων ἄρα καὶ ἡ ΖΒ τῆς ΖΗ ἡ ἐλάττων τῆς μείζονος· ὅπερ subtended by the greater side [Prop. 1.19]. Thus, FC (is)
ἐστὶν ἀδύνατον. οὐκ ἄρα ἡ ΖΗ κάθετός ἐστιν ἐπὶ τὴν ΔΕ. greater than FG. And FC (is) equal to FB. Thus, FB
ὁμοίως δὴ δεῖξομεν, ὅτι οὐδ᾿ ἄλλη τις πλὴν τῆς ΖΓ· ἡ ΖΓ (is) also greater than FG, the lesser than the greater. The
ἄρα κάθετός ἐστιν ἐπὶ τὴν ΔΕ. very thing is impossible. Thus, FG is not perpendicular to
ijþ (other) straight-line is joined from the center (of the cir-
᾿Εὰν ἄρα κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τοῦ DE. So, similarly, we can show that neither (is) any other
κέντρου ἐπὶ τὴν ἁφὴν ἐπιζευχθῇ τις εὐθεῖα, ἡ ἐπιζευχθεῖσα (straight-line) except FC. Thus, FC is perpendicular to
κάθετος ἔσται ἐπὶ τὴν ἐφαπτομένην· ὅπερ ἔδει δεῖξαι. DE. Thus, if some straight-line touches a circle, and some

cle) to the point of contact, then the (straight-line) so
joined will be perpendicular to the tangent. (Which is)
the very thing it was required to show.

Proposition 19
.
᾿Εὰν κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τῆς ἁφῆς τῇ If some straight-line touches a circle, and a straight-
ἐφαπτομένῃ πρὸς ὀρθὰς [γωνίας] εὐθεῖα γραμμὴ ἀχθῇ, ἐπὶ line is drawn from the point of contact, at right-[angles]
τῆς ἀχθείσης ἔσται τὸ κέντρον τοῦ κύκλου. to the tangent, then the center (of the circle) will be on
Κύκλου γὰρ τοῦ ΑΒΓ ἐφαπτέσθω τις εὐθεῖα ἡ ΔΕ κατὰ the (straight-line) so drawn.
τὸ Γ σημεῖον, καὶ ἀπὸ τοῦ Γ τῇ ΔΕ πρὸς ὀρθὰς ἤχθω ἡ For let some straight-line DE touch the circle ABC at
ΓΑ· λέγω, ὅτι ἐπὶ τῆς ΑΓ ἐστι τὸ κέντρον τοῦ κύκλου. point C. And let CA have been drawn from C, at right-

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