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Computable Euclid

Proposition 7a

Theorem

Let P be any point within a circle other than the center, and let PA , PB , PC , and PD  be lines to the circumference of the circle. If PA  passes through the center of the circle, PD  is the extension of PA  in the opposite direction, and PB  is nearer to PA  than PC  is to PA , then ￜPA ￜ > ￜPB ￜ > ￜPC ￜ > ￜPD ￜ.

Commentary

1. Given a circle centered at O, let P be any point inside the circle other than the center.
2. Connect PO  and extend it so that it intersects the circle at a point A. Extend AP  so that it intersects the circle at a point D.
3. Find two points B and C on the circumference, such that PB  is nearer to PA  than PC  is to PA . (While Euclid did not define "nearer" in this context, his proof suggests he had comparison of angles in mind, so that we are assuming ∠APB < ∠APC.)
4. Then ￜPA ￜ > ￜPB ￜ > ￜPC ￜ > ￜPD ￜ.
5. Book 3 Propositions 7b and 7c deal with the question of when line segments from P to the circumference can be equal in length.
6. Book 3 Proposition 8a is similar except P is outside of the circle.

Original statement

ἐὰν κύκλου ἐπὶ τῆς διαμέτρου ληϕθῇ τι σημϵῖον, ὃ μή ἐστι κέντρον τοῦ κύκλου, ἀπὸ δὲ τοῦ σημϵίου πρὸς τὸν κύκλον προσπίπτωσιν ϵὐθϵῖαί τινϵς, μϵγίστη μὲν ἔσται, ἐϕ᾽ ἧς τὸ κέντρον, ἐλαχίστη δὲ ἡ λοιπή, τῶν δὲ ἄλλων ἀϵὶ ἡ ἔγγιον τῆς διὰ τοῦ κέντρου τῆς ἀπώτϵρον μϵίζων ἐστίν, δύο δὲ μόνον ἴσαι ἀπὸ τοῦ σημϵίου προσπϵσοῦνται πρὸς τὸν κύκλον ἐϕ᾽ ἑκάτϵρα τῆς ἐλαχίστης.

English translation

If on the diameter of a circle a point is taken which is not the centre of the circle, and from the point straight lines fall upon the circle, that will be greatest on which the centre is, the remainder of the same diameter will be least, and of the rest the nearer to the straight line through the centre is always greater than the more remote, and only two equal straight lines will fall from the point on the circle, one on each side of the least straight line.

Computable version

Additional instances

Dependency graphs