48. Divide a given straight line into segments such that the dif ference of their squares may be equal to a given square. 49. Bisect a parallelogram by a straight line drawn through a given point. 50. Trisect a triangle by straight lines drawn from a given point in one of its sides. 51. Find a point in a given straight line such that the sum of the squares on its distances from two given points may be the least possible. 52. Find the locus of a point such that the sum of the squares on its distances from two given points may be equal to a given square. 53. Find the locus of a point such that the difference of the squares on its distances from two given points may be equal to a given square. DEFINITIONS OF BOOK II. DEF. I. The altitude of a parallelogram with reference to a given side as base is the perpendicular distance between the base and the opposite side. DEF. 2. The altitude of a triangle with reference to a given side as base is the perpendicular distance between the base and the opposite vertex. DEF. 3. The straight lines drawn through any point in a diagonal of a parallelogram parallel to the sides divide it into four parallelograms, of which the two whose diagonals are upon the given diagonal are called parallelograms about that diagonal, and the other two are called the complements of the parallelograms about the diagonal. DEF. 4. All rectangles being identically equal which have two adjoining sides equal to two given straight lines, any such rectangle is spoken of as the rectangle contained by those lines. In like manner, any square whose side is equal to a given straight line is spoken of as the square on that line. DEF 5. A point in a finite straight line is said to divide it internally, or, simply, to divide it; and by analogy, a point in the line produced is said to divide it externally; and, in either case, the distances of the point from the extremities of the line are called the segments of the line. Woodfall and Kinder, Printers, Milford Lane, Strand, London, W.C. |