but BAD is (Constr.) a right angle; therefore also (Ax. 3.) 5. ADE is a right angle; but the opposite angles of parallelograms are equal (I. 34.); therefore 6. Each of the opposite angles ABE, BED, is a right angle; and it has been demonstrated that it is equilateral; therefore (Def. 30.) ADEB is a square, 8. and it is described upon the given straight line AB. Q.E.F. COR.-Hence every parallelogram that has one right angle has all its angles right angles. PROP. XLVII.—THEOREM. In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle. Let ABC be a right-angled triangle having the right angle BAC; the square described the side BC is equal to the upon described squares BA, AC. upon On BC describe (I. 46.) the square BDEC, and on BA, AC, the squares GB, HC; through A draw (I. 31.) AL parallel to BD, or CE, and join AD, FC. Then, because each of the angles BAC, BAG, is a right angle (Def. 30.), the two straight lines AC, AG, upon the opposite sides of AB, make with it at the point 4 the adjacent angles equal to two right angles; therefore (I. 14.) 1. CA is in the same straight line with AG. For the same reason 2. AB and AH are in the same straight line. And because the angle DBC is equal to the angle FBA, each of them being a right angle, add to each the angle ABC, and (Ax. 2.) 3. The whole angle DBA is equal to the whole FBC. And because the two sides AB, BD, are equal to the two FB, BC, each to each, and the angle DBA equal to the angle FBC; therefore (I. 4.) 4. The base AD is equal to the base FC, and the triangle ABD to the triangle FBC. Now (I. 41.) 5. The parallelogram BL is double of the triangle ABD, because they are upon the same base BD, and between the same parallels BD, AL; and 6. The square GB is double of the triangle FBC, because these also are upon the same base FB, and between the same parallels FB, GC. But the doubles of equals are equal (Ax. 6.) to one another; therefore 7. The parallelogram BL is equal to the square GB. And, in the same manner, by joining AE, BK, it is demonstrated that 8. The parallelogram CL is equal to the square HC. Therefore (Ax. 2.) 9. The whole square BDEC is equal to the two squares GB, HC. And the square BDEC is described upon the straight line BC, and the squares GB, HC, upon BA, AC; wherefore 10. The square upon the side BC is equal to the squares upon the sides BA, AC. Therefore, in any right-angled triangle, &c. Q.E.D. If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle. If the square described upon BC, one of the sides of the triangle ABC, be equal to the squares upon the other sides BA, AC, the angle BAC is a right angle. D From the point A draw (I. 11.) AD at right angles to AC, and make AD equal to BA, and join DC. Then because DA is equal to AB, 1. The square of DA is equal to the square of AB. To each of these add the square of AC; therefore 2. The squares of DA, AC, are equal to the squares of But (I. 47.) 3. The square of DC is equal to the squares of DA, AC, because DAC is a right angle; and the square of BC, by hypothesis, is equal to the squares of BA, AC; therefore 4. The square of DC is equal to the square of BC; and therefore also 5. The side DC is equal to the side BC. And because the side DA is equal to AB, and AC common to the two triangles DAC, BAC, the two DA, AC, are equal to the two BA, AC, each to each, and the base DC is equal to the base BC; therefore (I. 8.) 6. The angle DAC is equal to the angle BAC. But DAC is (Constr.) a right angle; therefore also 7. BAC is a right angle. EUCLID'S ELEMENTS OF GEOMETRY. Book II. DEFINITIONS. I. Every right-angled parallelogram, or rectangle, is said to be contained by any two of the straight lines which contain one of the right angles. II. In every parallelogram, any of the parallelograms about a diameter, together with the two complements, is called a gnomon. "Thus the parallelogram HG, together with the complements AF, FC, is the gnomon, which is more briefly expressed by the letters 4GK, or EHC, which are at the opposite angles of the parallelograms which make the gnomon." PROP. I.-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 undivided line, and the several parts of the divided line. Let 4 and BC be two straight lines; and let BC be divided into any parts in the points D, E; the rectangle contained by the straight lines A, BC, is equal to the rectangle contained by A, BD, together with that contained by 4, DE, and that contained by A, EC. From the point B draw (I. 11.) BF at right angles to BC, and make BG equal (1.3.) to 4; and through G draw (I. 31.) GH parallel to BC; and through D, E, C, draw (I. 31.) DK, EL, CH, parallel to BG; then the rectangle BH is equal to the rectangles BK, DL, EH; and 1. BH is contained by A, BC, for it is contained by GB, BC, and GB is equal (Constr.) to 4; and 2. BK is contained by A, BD, for it is contained by GB, BD, of which GB is equal to 4; and 3. DL is contained by A, DE, because DK, that is, BG, (I. 34.) is equal to A; and in like manner 4. The rectangle EH is contained by A, EC. Therefore the rectangle contained by A, BC, is equal to the several rectangles contained by A, BD, by A, DE, and by A, EC. Wherefore, if there be two straight lines, &c. Q.E.D. PROP. II.-THEOREM. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square of the whole line. Let the straight line AB be divided into any two parts in the point C; the rectangle contained by AB, BC, together with the rectangle* AB, AC, shall be equal to the square of AB. Upon AB describe (I. 46.) the square ADEB,, and through C draw N.B. To avoid repeating the word contained too frequently, the rectangle contained by two straight lines AB, AC, is sometimes simply called the rectangle AB, AC. |