Let str. line AB be divided into any two parts in C, then AB2= AC + CB2 + 2. AC. CB. CGF || AD or BE, {HGK AB or DE; BD meets ||s AD, CF, Prop. 38. .. ext. ▲ BGC = int. oppo. ADB; Prop. 28. BA = AD, ▲ ABD = / ADB ; CBG BGC, = BC = CG, BC= GK, CG = GK; BC: = CG = GK = BK, fig. CK is equilat. CB meets ||s CG, BK, Prop. 5. Ax. 1. Prop. 6. Prop. 33. Ax. 1. KBC + ≤ GCB = 2 rt. Zs; Prop. 28. but KBC= rt. L .. fig. CK = CB2. In the same way it may be proved, that fig. AG= fig. GE; i.e. fig. AGAC.CG AC.CB. Prop. 33. Prop. 37. Ax. 1. fig. GE =AC.CB .. fig. AG + fig. GE= 2 AC.CB; AB2. = JAC2 + CB2 + 2. AC. CB. Wherefore, if a str. line, &c. COR. The likewise squares. ms about the dia. of a sq. are Note. If the parts of the divided line be a and b, the proposition algebraically expressed is (a+b)2=a2+2ab+b2. PROP. XLIV. THEOR. 5. 2 Eu. 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 whole line. Let the str. line AB be divided into two equal parts at the pt. C, and into two unequal parts at the point D; then AD. DB + CD2 = CB2. = fig. AL fig. DF, add to each the fig. CH, .. whole fig. AH = fig. DF+fig. CH, but fig. AH ={ and fig. DF + fig. CH: Prop. 43. JAD. DB, for DH Cor. .. gnomon CMG = AD. DB, add to each the fig. LG, which is CD3. .. gnomon CMG+fig. LG = AD. DB+CD2, i. e. whole fig. CEFB or CB2=AD.DB+CD2. Wherefore, if a str. line, &c. = COR.-Since CB2-CD2— AD.DB; the diff. of the Squares of two unequal lines is equal to the rectangle of their sum and diff. Note. If the unequal parts of the divided line be b and c, and the half line be a; the proposition algebraically expressed is be + (a - b)2=a2; or be + (c-a)2 = a2. Cor. Prop. 43. PROP. XLV. THEOR. 6. 2 Eu. If a straight line be bisected, and produced to E 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 line and the part produced. Let the str. line AB be bisected in C, and produced to D; then AD. DB + CB2 =CD'. Нур Prop. 35. Prop. 37. but, Cor. Prop. 43. join DE. BHG KLHM AK CE or DF, AD or EF, CL or DM; fig. AL= fig. CH; fig. CH= fig. HF, fig. AL fig. HF, add to each, the fig. CM. .. whole fig. AM = gnomon CMG, but fig. AM= AD.DB, for DM = DB; .. gnomon CMG = AD. DB, add to each, the fig. LG, which is equal to CB'. .. gnomon CMG + fig. LG = AD.DB+CB2, i. e. whole fig. CEFD or CD2 = AD.DB+CB2. Wherefore, if a str. line, &c. Note. If the half line be a, and the part produced b, the proposition algebraically expressed is (2a+b)b+a2=(a+b)2. PROP. XLVI. THEOR. 7. 2 Eu. 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 str. line AB be divided into any two parts at the pt. C; then AB+BC2=2.AB.BC + AC2. { CGF AD or BE, Prop. 30. HGK AB or DE; Prop. 37. Since fig. AG= fig. GE, add to each the fig. CK. .. fig. AK+fig. CE 2 fig. AK = but, fig. AK + fig. CE= gnomon AKF + .. = gnomon AKF+ fig. CK 2 2. fig. AK, fig. AK, for Cor. Prop. 43. |