PROP. XX. (xx.) If two chords of a given circle intersect each other, the angle of their inclination is the half of the angle at the center standing upon the aggregate, or the difference, of the arches intercepted between them, accordingly, as they meet within, or without the circle. (XXI.) To divide a given circular arch into two parts, so that the aggregate of their chords may be equal to a given straight line, greater than the chord of the whole arch, but not greater than the double of the chord of half the arch. (XXII.) To divide a given circular arch into two parts, so that the excess of the chord of the one above the chord of the other, may be equal to a given straight line, less than the chord of the whole. arch. (xxIII.) If from any given point, in the circumference of a circle, two straight lines be drawn to the ex tremities of a given chord, the angle which the one makes with any perpendicular to the chord, shall be equal to the angle which the other makes with the diameter of the circle that passes through the given point. PROP. XXI. (XXIV.) The perpendiculars let fall from the three angles of any triangle upon the opposite sides, intersect each other in the same point. COR. This point is equidistant from the three straight lines joining the points in which the perpendiculars meet the three sides of the triangle. (xxv.) If from a given point within a circle, which is not the center, straight lines be drawn to the circumference, making with each other equal angles, the two, which are nearer to the diameter passing through the given point, shall cut off a greater circumference than the two, which are more re mote. (xxvI.) From either of the two given points, in which two given circles intersect each other, to draw a chord cutting the one circumference, and meeting the other, such that the part of it, contained between the two circumferences, shall be equal to a given finite straight line. PROP. XXII. (XXVII.) If two opposite angles of a trapezium be together equal to two right angles, a circle may be described about it. (XXVIII.) A circle cannot be described about a rhombus, nor about any other parallelogram which is not rectangular. (XXIX.) If from any point, in the circumference of a given circle, straight lines be drawn to the three angles of an inscribed equilateral triangle, the greatest of them shall be equal to the aggregate of the two less. (xxx.) The first, third, fifth, &c. angles of any polygon, of an even number of sides, which is inscribed in a given circle, are together equal to the remaining angles of the figure; any angle whatever being as sumed as the first.* (XXXI.) To make a trapezium, about which a circle may be described, having its four sides respectively equal to four given straight lines, two of which are equal to each other, and any three together greater than the fourth; the two equal sides of the trapezium, also, being opposite to each other. * The Proposition may be adapted to the case of a polygon of an odd number of sides inscribed in a circle, by dividing any one of its angles into two, by a straight line drawn from the center, and reckoning the two segments of that angle, each as one of the angles of the figure; the number of angles is thus made even. The converse of this Theorem, and the second deduction from Prop. 18, may be applied to discover the relation which the angles of a polygon must have in order that it may admit of a circle being described about it, or inscribed in it. PROP. XXIV. (xxxII.) If upon the two greater sides of an oblong, as diameters, two semi-circles be described, lying to ward the same parts, the figure contained by the two remaining sides of the oblong, and the two curve lines, shall be equal to the oblong. PROP. XXVI. (xxxIII.) The straight lines joining the extremities of the chords of two equal arches of the same circle, toward the same parts, are parallel to each other. (XXXIV.) The arches of a circle that are intercepted between two parallel chords are equal to one another. (xxxv.) In equal circles the greater angle stands upon the greater circumference; whether the angles compared be at the centers or circumferences. |