Humanitext Reader

Euclid · Elements §3.prop.11-3.prop.13

Line Joining the Centers and Point of Contact of Circles

Passage 42 of 316 · Greek

Summary

Proves that the straight line joining the centers of two touching circles (whether internally or externally) passes through their point of contact, and demonstrates that two circles cannot touch each other at more than one point.

§3.prop.11ἐὰν δύο κύκλοι ἐφάπτωνται ἀλλήλων ἐντός, καὶ ληφθῇ αὐτῶν τὰ κέντρα, ἡ ἐπὶ τὰ κέντρα αὐτῶν ἐπιζευγνυμένη εὐθεῖα καὶ ἐκβαλλομένη ἐπὶ τὴν συναφὴν πεσεῖται τῶν κύκλων.
If two circles touch one another internally, and their centers be taken, the straight line joining their centers, being produced, will fall on the point of contact of the circles.
δύο γὰρ κύκλοι οἱ ΑΒΓ, ΑΔΕ ἐφαπτέσθωσαν ἀλλήλων ἐντὸς κατὰ τὸ Α σημεῖον, καὶ εἰλήφθω τοῦ μὲν ΑΒΓ κύκλου κέντρον τὸ Ζ, τοῦ δὲ ΑΔΕ τὸ Η· λέγω, ὅτι ἡ ἀπὸ τοῦ Η ἐπὶ τὸ Ζ ἐπιζευγνυμένη εὐθεῖα ἐκβαλλομένη ἐπὶ τὸ Α πεσεῖται.
For let two circles ΑΒΓ, ΑΔΕ touch one another internally at the point Α, and let the center Ζ of the circle ΑΒΓ, and the center Η of the circle ΑΔΕ, be taken; I say that the straight line joined from Η to Ζ, being produced, will fall on Α.
μὴ γάρ, ἀλλʼ εἰ δυνατόν, πιπτέτω ὡς ἡ ΖΗΘ, καὶ ἐπεζεύχθωσαν αἱ ΑΖ, ΑΗ. ἐπεὶ οὖν αἱ ΑΗ, ΗΖ τῆς ΖΑ, τουτέστι τῆς ΖΘ, μείζονές εἰσιν, κοινὴ ἀφῃρήσθω ἡ ΖΗ· λοιπὴ ἄρα ἡ ΑΗ λοιπῆς τῆς ΗΘ μείζων ἐστίν.
For if not, but if possible, let it fall as ΖΗΘ, and let ΑΖ, ΑΗ be joined. Since therefore ΑΗ, ΗΖ are greater than ΖΑ, that is, than ΖΘ, let ΖΗ be subtracted as common; therefore the remainder ΑΗ is greater than the remainder ΗΘ.
ἴση δὲ ἡ ΑΗ τῇ ΗΔ· καὶ ἡ ΗΔ ἄρα τῆς ΗΘ μείζων ἐστὶν ἡ ἐλάττων τῆς μείζονος· ὅπερ ἐστὶν ἀδύνατον·
But ΑΗ is equal to ΗΔ; therefore ΗΔ is also greater than ΗΘ, the less than the greater; which is impossible.
οὐκ ἄρα ἡ ἀπὸ τοῦ Ζ ἐπὶ τὸ Η ἐπιζευγνυμένη εὐθεῖα ἐκτὸς πεσεῖται· κατὰ τὸ Α ἄρα ἐπὶ τῆς συναφῆς πεσεῖται.
Therefore the straight line joined from Ζ to Η will not fall outside; therefore it will fall on the point of contact at Α.
ἐὰν ἄρα δύο κύκλοι ἐφάπτωνται ἀλλήλων ἐντός,, ἡ ἐπὶ τὰ κέντρα αὐτῶν ἐπιζευγνυμένη εὐθεῖα ἐπὶ τὴν συναφὴν πεσεῖται τῶν κύκλων· ὅπερ ἔδει δεῖξαι.
If therefore two circles touch one another internally, the straight line joining their centers will fall on the point of contact of the circles; which was to be proved.
§3.prop.12ἐὰν δύο κύκλοι ἐφάπτωνται ἀλλήλων ἐκτός, ἡ ἐπὶ τὰ κέντρα αὐτῶν ἐπιζευγνυμένη διὰ τῆς ἐπαφῆς ἐλεύσεται.
If two circles touch one another externally, the straight line joining their centers will pass through the point of contact.
δύο γὰρ κύκλοι οἱ ΑΒΓ, ΑΔΕ ἐφαπτέσθωσαν ἀλλήλων ἐκτὸς κατὰ τὸ Α σημεῖον, καὶ εἰλήφθω τοῦ μὲν ΑΒΓ κέντρον τὸ Ζ, τοῦ δὲ ΑΔΕ τὸ Η· λέγω, ὅτι ἡ ἀπὸ τοῦ Ζ ἐπὶ τὸ Η ἐπιζευγνυμένη εὐθεῖα διὰ τῆς κατὰ τὸ Α ἐπαφῆς ἐλεύσεται.
For let two circles ΑΒΓ, ΑΔΕ touch one another externally at the point Α, and let the center Ζ of the circle ΑΒΓ, and the center Η of the circle ΑΔΕ, be taken; I say that the straight line joined from Ζ to Η will pass through the point of contact at Α.
μὴ γάρ, ἀλλʼ εἰ δυνατόν, ἐρχέσθω ὡς ἡ ΖΓΔΗ, καὶ ἐπεζεύχθωσαν αἱ ΑΖ, ΑΗ. ἐπεὶ οὖν τὸ Ζ σημεῖον κέντρον ἐστὶ τοῦ ΑΒΓ κύκλου, ἴση ἐστὶν ἡ ΖΑ τῇ ΖΓ. πάλιν, ἐπεὶ τὸ Η σημεῖον κέντρον ἐστὶ τοῦ ΑΔΕ κύκλου, ἴση ἐστὶν ἡ ΗΑ τῇ ΗΔ. ἐδείχθη δὲ καὶ ἡ ΖΑ τῇ ΖΓ ἴση·
For if not, but if possible, let it go as ΖΓΔΗ, and let ΑΖ, ΑΗ be joined. Since therefore the point Ζ is the center of the circle ΑΒΓ, ΖΑ is equal to ΖΓ. Again, since the point Η is the center of the circle ΑΔΕ, ΗΑ is equal to ΗΔ.
αἱ ἄρα ΖΑ, ΑΗ ταῖς ΖΓ, ΗΔ ἴσαι εἰσίν· ὥστε ὅλη ἡ ΖΗ τῶν ΖΑ, ΑΗ μείζων ἐστίν· ἀλλὰ καὶ ἐλάττων· ὅπερ ἐστὶν ἀδύνατον.
But ΖΑ was also proved equal to ΖΓ; therefore the two ΖΑ, ΑΗ are equal to the two ΖΓ, ΗΔ; so that the whole ΖΗ is greater than ΖΑ, ΑΗ; but it is also less; which is impossible.
οὐκ ἄρα ἡ ἀπὸ τοῦ Ζ ἐπὶ τὸ Η ἐπιζευγνυμένη εὐθεῖα διὰ τῆς κατὰ τὸ Α ἐπαφῆς οὐκ ἐλεύσεται· διʼ αὐτῆς ἄρα.
Therefore the straight line joined from Ζ to Η will not fail to pass through the point of contact at Α; therefore it will pass through it.
ἐὰν ἄρα δύο κύκλοι ἐφάπτωνται ἀλλήλων ἐκτός, ἡ ἐπὶ τὰ κέντρα αὐτῶν ἐπιζευγνυμένη διὰ τῆς ἐπαφῆς ἐλεύσεται· ὅπερ ἔδει δεῖξαι.
If therefore two circles touch one another externally, the straight line joining their centers will pass through the point of contact; which was to be proved.
§3.prop.13κύκλος κύκλου οὐκ ἐφάπτεται κατὰ πλείονα σημεῖα ἢ καθʼ ἕν, ἐάν τε ἐντὸς ἐάν τε ἐκτὸς ἐφάπτηται.
A circle does not touch a circle at more points than one, whether it touch it internally or externally.
εἰ γὰρ δυνατόν, κύκλος ὁ ΑΒΓΔ κύκλου τοῦ ΕΒΖΔ ἐφαπτέσθω πρότερον ἐντὸς κατὰ πλείονα σημεῖα ἢ ἓν τὰ δ, Β. καὶ εἰλήφθω τοῦ μὲν ΑΒΓΔ κύκλου κέντρον τὸ Η, τοῦ δὲ ΕΒΖΔ τὸ Θ. ἡ ἄρα ἀπὸ τοῦ Η ἐπὶ τὸ Θ ἐπιζευγνυμένη ἐπὶ τὰ Β, Δ πεσεῖται.
For, if possible, let a circle ΑΒΓΔ touch a circle ΕΒΖΔ first internally at more points than one, namely at δ, Β. And let the center Η of the circle ΑΒΓΔ, and the center Θ of the circle ΕΒΖΔ, be taken. Therefore the straight line joined from Η to Θ will fall on Β, Δ.
πιπτέτω ὡς ἡ ΒΗΘΔ. καὶ ἐπεὶ τὸ Η σημεῖον κέντρον ἐστὶ τοῦ ΑΒΓΔ κύκλου, ἴση ἐστὶν ἡ ΒΗ τῇ ΗΔ· μείζων ἄρα ἡ ΒΗ τῆς ΘΔ· πολλῷ ἄρα μείζων ἡ ΒΘ τῆς ΘΔ. πάλιν, ἐπεὶ τὸ Θ σημεῖον κέντρον ἐστὶ τοῦ ΕΒΖΔ κύκλου, ἴση ἐστὶν ἡ ΒΘ τῇ ΘΔ· ἐδείχθη δὲ αὐτῆς καὶ πολλῷ μείζων· ὅπερ ἀδύνατον·
Let it fall as ΒΗΘΔ. And since the point Η is the center of the circle ΑΒΓΔ, ΒΗ is equal to ΗΔ; therefore ΒΗ is greater than ΘΔ; therefore ΒΘ is much greater than ΘΔ. Again, since the point Θ is the center of the circle ΕΒΖΔ, ΒΘ is equal to ΘΔ; but it was also proved to be much greater than it; which is impossible.
οὐκ ἄρα κύκλος κύκλου ἐφάπτεται ἐντὸς κατὰ πλείονα σημεῖα ἢ ἕν.
Therefore a circle does not touch a circle internally at more points than one.
λέγω δή, ὅτι οὐδὲ ἐκτός.
I say then that neither does it touch externally.
εἰ γὰρ δυνατόν, κύκλος ὁ ΑΓΚ κύκλου τοῦ ΑΒΓΔ ἐφαπτέσθω ἐκτὸς κατὰ πλείονα σημεῖα ἢ ἓν τὰ Α, Γ, καὶ ἐπεζεύχθω ἡ ΑΓ. ἐπεὶ οὖν κύκλων τῶν ΑΒΓΔ, ΑΓΚ εἴληπται ἐπὶ τῆς περιφερείας ἑκατέρου δύο τυχόντα σημεῖα τὰ Α, Γ, ἡ ἐπὶ τὰ σημεῖα ἐπιζευγνυμένη εὐθεῖα ἐντὸς ἑκατέρου πεσεῖται· ἀλλὰ τοῦ μὲν ΑΒΓΔ ἐντὸς ἔπεσεν, τοῦ δὲ ΑΓΚ ἐκτός· ὅπερ ἄτοπον·
For, if possible, let a circle ΑΓΚ touch a circle ΑΒΓΔ externally at more points than one, namely Α, Γ, and let ΑΓ be joined. Since therefore on the circumference of each of the circles ΑΒΓΔ, ΑΓΚ two random points Α, Γ have been taken, the straight line joining the points will fall inside each of them; but it fell inside ΑΒΓΔ, and outside ΑΓΚ; which is absurd.
οὐκ ἄρα κύκλος κύκλου ἐφάπτεται ἐκτὸς κατὰ πλείονα σημεῖα ἢ ἕν.
Therefore a circle does not touch a circle externally at more points than one.
ἐδείχθη δέ, ὅτι οὐδὲ ἐντός.
And it was also proved that neither does it internally.
κύκλος ἄρα κύκλου οὐκ ἐφάπτεται κατὰ πλείονα σημεῖα ἢ ἕν, ἐάν τε ἐντὸς ἐάν τε ἐκτὸς ἐφάπτηται· ὅπερ ἔδει δεῖξαι.
Therefore a circle does not touch a circle at more points than one, whether it touch it internally or externally; which was to be proved.

Notes

  1. §3.prop.11μὴ γάρ, ἀλλʼ εἰ δυνατόν — A formulaic expression in geometrical proofs introducing a proof by contradiction. It is an elliptical expression meaning 'For let it not be so, but, if possible, let it be...'.
  2. §3.prop.11ἡ ἐλάττων τῆς μείζονος — An appositional or predicative phrase expressing a contradiction: 'the less [being greater] than the greater'. It highlights the geometrical absurdity under the assumed contradiction where a part (radius ΗΔ) is declared to be greater than the whole segment (ΗΘ).
  3. §3.prop.12οὐκ ἄρα ἡ ἀπὸ τοῦ Ζ ἐπὶ τὸ Η ἐπιζευγνυμένη εὐθεῖα διὰ τῆς κατὰ τὸ Α ἐπαφῆς οὐκ ἐλεύσεται — The negative particle οὐκ at the beginning of the clause and οὐκ before the verb forming a double negation, resulting in a strong affirmation: 'it will not fail to pass...' (i.e., it must pass through). This is used to negate the false assumption in the conclusion of the reductio ad absurdum.
  4. §3.prop.13ἡ ἐπὶ τὰ σημεῖα ἐπιζευγνυμένη εὐθεῖα ἐντὸς ἑκατέρου πεσεῖται — Based on the previously proved theorem (Book III, Proposition 2) that a straight line joining two points on the circumference of a circle falls inside it. Under the assumption that the two circles touch externally, this leads to the absurd physical contradiction that the joining line must fall inside both circles simultaneously.

Cite this passage

Euclid, Elements §3.prop.11-3.prop.13. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:3.prop.11-3.prop.13

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