(xxxI.) If either angle at the base of a triangle be a right angle, and if the base be divided into any number of equal parts, that which is adjacent to the right angle shall subtend the greatest angle at the vertex; and, of the rest, that which is nearer to the right angle shall subtend, at the vertex, a greater angle than that which is more remote. (xxxII.) To trisect a given finite straight line. COR. Hence, to inscribe a square in a given right-angled isosceles triangle. (xxxIII.) To describe a triangle which shall have its three sides, taken together, equal to a given finite straight line, and its three angles equal to three given angles, each to each; the three given angles. being together equal to two right angles. (XXXIV.) If, in the sides of a given square, at equal distances from the four angular points, four other points be taken, one in each side, the figure contained by the straight lines which join them, shall also be a square. (XXXV.) If the opposite angles, of a quadrilateral figure be equal to each other, the figure shall be a parallelogram. (XXXVI.) In a given square to inscribe an equilateral triangle, having one of its angular points upon one of the angular points of the square, and its two remaining angular points one in each of two adjacent sides of the square. (XXXVII.) If, at the extremities of the base of a given triangle, two straight lines be drawn, both above the base, and each of them equal to the adjacent side, and making with it an angle equal to the vertical angle of the triangle; then, if two straight lines, let fall from the extremities of the two so drawn, make, with the base produced, two angles that are equal each of them to the vertical angle, they shall cut off equal segments from the base produced. (XXXVIII.) To inscribe a square in a given rhombus. (XXXIX.) If four straight lines cut each other, without including space, but so as to make three internal angles, towards the same parts, which together are less than four right angles, the two lines, which are not joined, shall meet, if produced far enough. (XL.) If the straight line, drawn from a point in the produced diameter of a circle to the convex circumference be equal to the half of the diameter, the angle, at the center, subtended by the concave circumference included between the diameter and the line so drawn, is the triple of the angle, at the center, subtended by the convex circumference included between the same two lines, The converse of the proposition is also true. COR. Hence, if a straight line could be drawn from any point in the curve of a semi-circle to meet the diameter produced, so that the part of the line without the curve should be equal to the radius, any angle might be trisected. One of the two sides, which are about the right angle of a right-angled triangle, and the aggregate of the hypotenuse and the remaining side, being given, to construct the triangle. PROP. XXXIV. (XLII.) The diameters of a parallelogram bisect each other. (XLIII.) The diameters of an equilateral four-sided plane rectilineal figure bisect one another at right angles. (XLIV.) The diameters of a rectangle are equal to one another. (XLV.) If two opposite sides of a parallelogram be divided each into the same number of equal parts, the straight lines, joining the opposite points of division, shall also divide the diameter of the parallelogram into the same number of equal parts. (XLVI.) To divide a given finite straight line into any given number of equal parts. (XLVII.) Upon a given finite straight line, as a diameter, to describe a square. (XLVIII.) Upon a given finite straight line to describe an equilateral and equiangular octagon. (XLIX.) If either diameter of a parallelogram be equal to a side of the figure, the other diameter shall be greater than any side of the figure. (L.) From a given point to draw a straight line. cutting two parallel straight lines, so that the part of it, intercepted between them, shall be equal to a given finite straight line, not less than the perpendicular distance of the two parallels. (LI.) If, from the summit of the right angle of a scalene right-angled triangle, two straight lines be drawn, one perpendicular to the hypotenuse, and the other bisecting it, they shall contain an angle equal to the difference of the two acute angles of the triangle. |