Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/77

Rh

Thus the parallelogram HG, together with the complements AF,FC, is the gnomon, which is more briefly expressed by the letters AGK, or EHC, which are at the opposite angles of the parallelograms which make the gnomon.

PROPOSITION 1. THEOREM.

If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undecided line, and the several parts of the divided line.

Let A and BC be two straight lines; and let BC be divided into any number of parts at the points D, E: the rectangle contained by the straight lines A, BC, shall be equal to the rectangle contained by A, BD, together with that contained by A, DE, and that contained by A, EC.

From the point B draw BF at right angles to BC; [I. 11. and make BG equal to A; [I. 3. through G draw GH parallel to BC;, and through D, E, C draw DK, EL, CH, parallel to BG. [I. 31.

Then the rectangle BH is equal to the rectangles BK, DL, EH. But BH is contained by A, BC, for it is contained by GB, BC, and GB is equal to A. [Construction. And BK is contained by A, BD, for it is contained by GB, BD, and GB is equal to A; and DL is contained by A, DE, because DK is equal to BG, which is equal to A; [I. 34. and in like manner EH is contained by A, EC. Therefore the rectangle contained by A, BC is, equal to the rectangles contained by A,BD, and by A, DE, and by A, EC. Wherefore, if there be two straight lines &c.