Bibliotheca Polyglotta

If a point be taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut the circle, and the other fall on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference be equal to the square on the straight line which falls on the circle, the straight line which falls on it will touch the circle.

SI extra circulum sumatur punctum aliquod, ab eoque puncto in circulum cadant duæ rectæ lineæ, quarum altera circulum secet, altera in eum incidat; sit autem, quod sub tota secante, & exterius inter punctum, & conuexam peripheriam assumpta, comprehenditur rectangulum, æquale ei, quod ab incidente describitur, quadrato; Incidens ipsa circulum tanget.

圜外任於一點、出兩直線。一至規外。一割圜、至規內。而割圜全線、偕割圜之規外線、矩內直角形。與至規外之線上直角方形、等。則至規外之線必切圜。

For let a point D be taken outside the circle ABC; from D let the two straight lines DCA, DB fall on the circle ACB; let DCA cut the circle and DB fall on it; and let the rectangle AD, DC be equal to the square on DB. I say that DB touches the circle ABC.

For let DE be drawn touching ABC; let the centre of the circle ABC be taken, and let it be F; let FE, FB, FD be joined. Thus the angle FED is right. [III. 18] Now, since DE touches the circle ABC, and DCA cuts it, the rectangle AD, DC is equal to the square on DE. [III. 36] But the rectangle AD, DC was also equal to the square on DB; therefore the square on DE is equal to the square on DB; therefore DE is equal to DB. And FE is equal to FB; therefore the two sides DE, EF are equal to the two sides DB, BF; and FD is the common base of the triangles; therefore the angle DEF is equal to the angle DBF. [I. 8] But the angle DEF is right; therefore the angle DBF is also right. And FB produced is a diameter; and the straight line drawn at right angles to the diameter of a circle, from its extremity, touches the circle; [III. 16, Por.] therefore DB touches the circle. Similarly this can be proved to be the case even if the centre be on AC.

Therefore etc. Q. E. D.

BOOK IV.

EVCLIDIS ELEMENTVM QVARTVM.

幾本原本第四卷之首

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