angle 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 A and BC be two str. lines, and let BC be divided into any parts in D and E; then A.BC=A. BD+A. DE+A. EC B D E C PROP. XLII. THEOR. 3. 2 Eu. If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rect angle contained by the two parts, together with the square of the aforesaid part. Let the str. line AB be divided into two parts in the pt. C; then AB. BC = AC. BC + BC. A C B PROP. XLIII. THEOR. 4. 2 Eu. If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. 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 + CD = CB. C D B 17 EG F |