Bibliotheca Polyglotta

If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

SI fuerint duæ rectæ lineæ, seceturque ipsarum altera in quotcunque segmenta: Rectangulum comprehensum sub illis duabus rectis lineis, æquale est eis, quæ sub insecta, & quolibet segmentorum comprehenduntur, rectangulis.

兩直線。任以一線、任分為若干分。其兩元線矩內直角形。與不分線、偕諸分線、矩內諸直角形幷、等。

Let A, BC be two straight lines, and let BC be cut at random at the points D, E; I say that the rectangle contained by A, BC is equal to the rectangle contained by A, BD, that contained by A, DE and that contained by A, EC.

For let BF be drawn from B at right angles to BC; [I. 11] let BG be made equal to A, [I. 3] through G let GH be drawn parallel to BC, [I. 31] and through D, E, C let DK, EL, CH be drawn parallel to BG.

Then BH is equal to BK, DL, EH. Now BH is the rectangle A, BC, for it is contained by GB, BC, and BG is equal to A; BK is the rectangle A, BD, for it is contained by GB, BD, and BG is equal to A; and DL is the rectangle A, DE, for DK, that is BG [I. 34], is equal to A. Similarly also EH is the rectangle A, EC. Therefore the rectangle A, BC is equal to the rectangle A, BD, the rectangle A, DE and the rectangle A, EC.

Therefore etc. Q. E. D.

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