Page 92 - Euclid's Elements of Geometry

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ELEMENTS BOOK 3

ΒΕΔ ἐστι διπλσίων· ἴση ἄρα ἡ ὑπὸ ΒΑΔ τῇ ὑπὸ ΒΕΔ. So, for the same (reasons), BFD is also double BED.
᾿Εν κύκλῳ ἄρα αἱ ἐν τῷ αὐτῷ τμήματι γωνίαι ἴσαι Thus, BAD (is) equal to BED.
Thus, in a circle, angles in the same segment are equal
ἀλλήλαις εἰσίν· ὅπερ ἔδει δεῖξαι. to one another. (Which is) the very thing it was required

to show.
Proposition 22
.
Τῶν ἐν τοῖς κύκλοις τετραπλεύρων αἱ ἀπεναντίον γωνίαι For quadrilaterals within circles, the (sum of the) op-
δυσὶν ὀρθαῖς ἴσαι εἰσίν. posite angles is equal to two right-angles.
Β B

Α A

Γ C

∆ D
῎Εστω κύκλος ὁ ΑΒΓΔ, καὶ ἐν αὐτῷ τετράπλευρον ἔστω Let ABCD be a circle, and let ABCD be a quadrilat-
τὸ ΑΒΓΔ· λέγω, ὅτι αἱ ἀπεναντίον γωνίαι δυσὶν ὀρθαῖς ἴσαι eral within it. I say that the (sum of the) opposite angles
εἰσίν. is equal to two right-angles.
᾿Επεζεύχθωσαν αἱ ΑΓ, ΒΔ. Let AC and BD have been joined.
᾿Επεὶ οὖν παντὸς τριγώνου αἱ τρεῖς γωνίαι δυσὶν ὀρθαῖς Therefore, since the three angles of any triangle are
ἴσαι εἰσίν, τοῦ ΑΒΓ ἄρα τριγώνου αἱ τρεῖς γωνίαι αἱ ὑπὸ equal to two right-angles [Prop. 1.32], the three angles
ΓΑΒ, ΑΒΓ, ΒΓΑ δυσὶν ὀρθαῖς ἴσαι εἰσίν. ἴση δὲ ἡ μὲν ὑπὸ CAB, ABC, and BCA of triangle ABC are thus equal
ΓΑΒ τῇ ὑπὸ ΒΔΓ· ἐν γὰρ τῷ αὐτῷ τμήματί εἰσι τῷ ΒΑΔΓ· to two right-angles. And CAB (is) equal to BDC. For
ἡ δὲ ὑπὸ ΑΓΒ τῇ ὑπὸ ΑΔΒ· ἐν γὰρ τῷ αὐτῷ τμήματί εἰσι they are in the same segment BADC [Prop. 3.21]. And
τῷ ΑΔΓΒ· ὅλη ἄρα ἡ ὑπὸ ΑΔΓ ταῖς ὑπὸ ΒΑΓ, ΑΓΒ ἴση ACB (is equal) to ADB. For they are in the same seg-
ἐστίν. κοινὴ προσκείσθω ἡ ὑπὸ ΑΒΓ· αἱ ἄρα ὑπὸ ΑΒΓ, ment ADCB [Prop. 3.21]. Thus, the whole of ADC is
ΒΑΓ, ΑΓΒ ταῖς ὑπὸ ΑΒΓ, ΑΔΓ ἴσαι εἰσίν. ἀλλ᾿ αἱ ὑπὸ equal to BAC and ACB. Let ABC have been added to
kgþ the) opposite angles is equal to two right-angles. (Which
ΑΒΓ, ΒΑΓ, ΑΓΒ δυσὶν ὀρθαῖς ἴσαι εἰσίν. καὶ αἱ ὑπὸ ΑΒΓ, both. Thus, ABC, BAC, and ACB are equal to ABC
ΑΔΓ ἄρα δυσὶν ὀρθαῖς ἴσαι εἰσίν. ὁμοίως δὴ δείξομεν, ὅτι and ADC. But, ABC, BAC, and ACB are equal to two
καὶ αἱ ὑπὸ ΒΑΔ, ΔΓΒ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν. right-angles. Thus, ABC and ADC are also equal to two
Τῶν ἄρα ἐν τοῖς κύκλοις τετραπλεύρων αἱ ἀπεναντίον right-angles. Similarly, we can show that angles BAD
γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν· ὅπερ ἔδει δεῖξαι. and DCB are also equal to two right-angles.
Thus, for quadrilaterals within circles, the (sum of
is) the very thing it was required to show.

.
Proposition 23
᾿Επὶ τῆς αὐτῆς εὐθείας δύο τμήματα κύκλων ὅμοια καὶ Two similar and unequal segments of circles cannot be
ἄνισα οὐ συσταθήσεται ἐπὶ τὰ αὐτὰ μέρη. constructed on the same side of the same straight-line.
Εἰ γὰρ δυνατόν, ἐπὶ τῆς αὐτῆς εὐθείας τῆς ΑΒ δύο For, if possible, let the two similar and unequal seg-
τμήματα κύκλων ὅμοια καὶ ἄνισα συνεστάτω ἐπὶ τὰ αὐτὰ ments of circles, ACB and ADB, have been constructed
μέρη τὰ ΑΓΒ, ΑΔΒ, καὶ διήχθω ἡ ΑΓΔ, καὶ ἐπεζεύχθωσαν on the same side of the same straight-line AB. And let

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