Upon AB describe the square ADEB, (1. 46.) join BD, and through C draw CGF parallel to AD or BE, (1.31.) and through G draw HGK parallel to AB or DE. Then, because CF is parallel to AD, and BD falls upon them, but the angle ADB is equal to the angle ABD, (1. 5.) for, since CG is parallel to BK, and CB meets them, therefore the angles KBC, GCB are equal to two right angles; (1. 29.) but the angle KBC is a right angle; (def. 30. constr.) wherefore GCB is a right angle: and therefore also the angles CGK, GKB, opposite to these, are right angles; (1. 34.) wherefore CGKB is rectangular : it is also equilateral, as was demonstrated; wherefore it is a square, and it is upon the side CB. For the same reason HF is a square, and it is upon the side HG, which is equal to AC. (1. 34.) Therefore the figures HF, CK, are the squares of AC, CB. And because the complement AG is equal to the complement GE, (1.43.) and that AG is the rectangle contained by AC, CB, for GC is equal to CB; therefore GE is also equal to the rectangle AC, CB; wherefore AG, GE are equal to twice the rectangle AC, CB; and HF, CK are the squares of AC, CB; wherefore the four figures HF, CK, AG, GE, are equal to the squares of AC, CB, and to twice the rectangle AC, CB: but HF, CK, AG, GE make up the whole figure ADEB, which is the square of AB; therefore the square of AB is equal to the squares of AC, CB, and twice the rectangle AC, CB. Wherefore, if a straight line be divided, &c. Q.E. D. COR. From the demonstration, it is manifest, that the parallelograms about the diameter of a square are likewise squares. PROPOSITION V. THEOREM. If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line. Let the straight line AB be divided into two equal parts in the point C, and into two unequal parts in the point D. Then the rectangle AD, DB, together with the square of CD, shall be equal to the square of CB. Upon CB describe the square CEFB, (1. 46.) join BE, and through D draw DHG parallel to CE or BF; (1.31.) and through H draw KLM parallel to CB or EF; also through A draw AK parallel to CL or BM. Then, because the complement CH is equal to the complement HF, (1. 43.). to each of these equals add DM; therefore the whole CM is equal to the whole DF; but because AC is equal to CB, therefore CM is equal to AL, (1. 36.) therefore also AL is equal to DF: to each of these equals add CH, and therefore the whole AH is equal to DF and CH: but AH is the rectangle contained by AD, DB, for DH is equal to DB; and DF together with CH is the gnomon CMG; therefore the gnomon CMG is equal to the rectangle AD, DB: to each of these equals add LG, which is equal to the square of CD; (11. 4. Cor.) therefore the gnomon CMG, together with LG, is equal to the rectangle ÅD, DB, together with the square of CD: but the gnomon CMG and LG make up the whole figure CEFB, which is the square of CB; therefore the rectangle AD, DB, together with the square of CD, is equal to the square of CB. Wherefore, if a straight line, &c. Q. E.D. COR. From this proposition it is manifest, that the difference of the squares of two unequal lines AC, CD, is equal to the rectangle contained by their sum AD and their difference DB. PROPOSITION VI. THEOREM. If a straight line be bisected, and produced to any point; the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line which is made up of the half and the part produced. Let the straight line AB be bisected in C, and produced to the point D. Then the rectangle AD, DB, together with the square of CB, shall be equal to the square of CD. Upon CD describe the square CEFD, (1. 46.) and join DE, therefore the rectangle AL is equal to the rectangle CH, (1. 36.) to each of these equals add CM; therefore the whole AM is equal to the gnomon CMG: therefore the gnomon CMG is equal to the rectangle AD, DB: add to each of these equals LG which is equal to the square of CB; therefore the rectangle AD, DB, together with the square of CB, is equal to the gnomon CMG, and the figure LG; but the gnomon CMG and LG make up the whole figure CEFD, which is the square of CD; therefore the rectangle AD, DB, together with the square of CB, is equal to the square of CD. Wherefore, if a straight line, &c. Q. E.D. If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts in the point C. Then the squares of AB, BC shall be equal to twice the rectangle, AB, BC, together with the square of AC. Upon AB describe the square ADEB, (1. 46.) and join BD; through C draw CF parallel to AD or BE cutting BD in G, (1. 31.) through G draw HGK parallel to AB or DE. Then because AG is equal to GE, (1.43.) add to each of them CK; therefore the whole AK is equal to the whole CE; but AK, CE are the gnomon AKF and the square CK; therefore the gnomon AKF and the square CK is double of AK: but twice the rectangle AB, BC, is double of AK, for BK is equal to BC; (II. 4. Cor.) therefore the gnomon AKF and the square CK, is equal to twice the rectangle AB, BC; square of to each of these equals add HF, which is equal to the but the gnomon AKF, together with the squares CK, HF, make up the whole figure ADEB and CK, which are the squares of AB and BC; therefore the squares of AB and BC are equal to twice the rectangle AB, BC, together with the square of AC. If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line, which is made up of the whole and that part. Let the straight line AB be divided into any two parts in the point C. Then four times the rectangle AB, BC, together with the square of AC, shall be equal to the square of the straight line made up of AB and BC together. Produce AB to D, so that BD be equal to CB, (1.3.) upon AD describe the square AEFD, (1. 46.) and join BE, through B, C, draw BL, CH parallel to AE or DF, and cutting DE in the points K, P respectively; through K, P, draw MGKN, XPRO parallel to AD or EF. Then because CB is equal to BD, CB to GK, and BD to KN; therefore GK is equal to KN; for the same reason, PR is equal to RO; and because CB is equal to BD, and GK to KN, therefore the rectangle CK is equal to BN, and GR to RN; (1.36.) but CK is equal to RN, (1. 43.) because they are the complements of the parallelogram CO; and therefore the four rectangles BÑ, CK, GR, RN, are equal to one another, and so are quadruple of one of them CK. Again, because CB is equal to BD, and BD to BK, that is, to CG; and because CB is equal to GK, that is, to GP; therefore CG is equal to GP. And because CG is equal to GP, and PR to RO, because they are the complements of the parallelogram ML; therefore the four rectangles AG, MP, PL, RF, are equal to one another, and so are quadruple of one of them AG. And it was demonstrated that the four CK, BN, GR, and RN are quadruple of CK; therefore the eight rectangles which contain the gnomon AOH, are quadruple of AK. And because AK is the rectangle contained by AB, BC, for BK is equal to BC; therefore four times the rectangle AB, BC is quadruple of AK; but the gnomon AOH was demonstrated to be quadruple of AK; therefore four times the rectangle AB, BC is equal to the gnomon AOH; to each of these equals add XH, which is equal to the square of AC; therefore four times the rectangle AB, BC, together with the square of AC, is equal to the gnomon AOH and the square XH; but the gnomon AOH and XH make up the figure AEFD which is the square of AD; therefore four times the rectangle AB, BC together with the square of AC, is equal to the square of AD, that is, of AB and BC added together is one straight line. Wherefore, if a straight line, &c. Q.E.D. PROPOSITION IX. THEOREM. If a straight line be divided into two equal, and also into two unequal parts; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section. Let the straight line AB be divided into two equal parts in the point C, and into two unequal parts in the point D. Then the squares of AD, DB together, shall be double of the squares of AC, CD. From the point C draw CE at right angles to AB, (1.11.) therefore the angle EAC is equal to the angle AEC; (1.5.) therefore the two other angles AEC, EAC of the triangle are and since they are equal to one another; therefore each of them is half of a right angle. For the same reason, each of the angles CEB, EBC is half a right angle ; and therefore the whole AEB is a right angle. And because the angle GEF is half a right angle, and EGF a right angle, for it is equal to the interior and opposite angle ECB, (1. 29.) |