204 ELEMENTS OF GEOMETRY. Book IV. difference of the sides. Again, because AD, drawn from the a 1. 2. centre, cuts GC at right angles, it also bisects it; therefore, when the perpendicular falls within the triangle BG = DGDB = DC-DB, the difference of the segments of the base; and BC BD+DC the sum of the segments. But when AD falls without the triangle BG = DG+ DB = CD+ DB, the sum of the segments of the base, and BC = CDDB, the difference of the segments of the base. Now, in both cases, because B is the intersection of the two lines FE, GC drawn in the circle, FB.BE = CB.BG, that is, as has been shown, (AC+AB).(AC — AB) = (CD+DB).(CD—DB). Therefore, &c. Q. E. D. EXERCISES ON GEOMETRY. BOOK I. 1. If a straight line bisect another at right angles, every point of the first line will be equally distant from the two extremities of the second line. 2. If two sides of a triangle be bisected at right angles, and from the point, where the bisecting lines cut one another, a perpendicular be drawn to the third side, it will bisect the third side. 3. If two angles of a triangle be bisected, and from the point where the bisecting lines cut one another, a straight line be drawn to the third angle, it will bisect the third angle. 4. Of all the straight lines that can be drawn from a point to a straight line, the perpendicular is the least; that which is nearer the perpendicular is less than the more remote; there can be two equal lines, but there cannot be more than two. 5. The difference of any two sides of a triangle is less than the third side. 6. If, from the vertex of a triangle, two straight lines be drawn, one of which is perpendicular to the base, and the other bisects the vertical angle, the angle contained by these lines shall be equal to half the difference of the angles at the base. 7. If a straight line bisect the diameter of a parallelogram, it will bisect the parallelogram. 8. If a straight line halve one side of a triangle, and be parallel to the base, it will halve the other side, and be equal to half the base; and the triangle cut off will be equal to one-fourth part of the original triangle. 9. If a straight line halve the two sides of an angle, it will be parallel to the base. 10. If a point be taken in each side of a square, at the same distance from the extremity, the figure formed by joining the four points will also be a square. 11. If, from the vertex of a triangle, two straight lines be drawn, cutting off from the base segments equal to the sides next them respectively, they shall contain an angle equal to half the sum of the angles at the base. 12. To construct a triangle, having the angles at the base equal to two given angles, (which are together less than the right angles,) each to each, and its perimeter, or the sum of all its sides, equal to a given straight line. BOOK II. 1. If, in a circle, a straight line bisect two parallel chords, it will pass through the centre. 2. Parallel chords intercept equal arcs. 3. Chords that intercept equal arcs, are parallel. 4. If two chords intersect at a point within the circle, the angle which they make is equal to an angle at the centre, standing on half the sum of the intercepted arcs. 5. If two chords intersect when produced, the angle which they make is equal to an angle at the centre, on half the difference of the intercepted arcs. 6. If two chords intersect at right angles, the sum of the two opposite arcs is equal to half the circumference. 7. A straight line drawn through the middle of an arc, parallel to its chord, is a tangent. 8. If two chords in a circle intersect at right angles, the sum of the squares of the four segments, will be equivalent to the square of the diameter. 9. If the circumference of a circle be divided into three equal parts, and from the three points straight lines be drawn to a fourth point in the circumference, one of the straight lines will be equal to the sum of the other two. BOOK III. 1. If, from the vertex of an isosceles triangle, a straight line be drawn to meet the base, the square of this line, together with the rectangle under the segments of the base, shall be equal to the square of one of the equal sides of the triangle. 2. The square of the base of an isosceles triangle, is equal to twice the rectangle contained by either side, and by the straight line intercepted between the perpendicular, let fall upon it from the opposite angle and the extremity of the base. 3. If the vertical angle of a triangle be two-thirds of a right angle, the square of the base will be equal to the dif ference between the rectangle under the two sides and the sum of their squares. 4. In any quadrilateral figure, the sum of the squares of the diagonals, together with four times the square of the line joining their middle points, is equal to the sum of the squares of all the sides. 5. In any quadrilateral figure, the sum of the squares of the diagonals, is equal to double the sum of the squares of the two lines, that join the middle points of its opposite sides. BOOK IV. 1. If two straight lines be cut by parallel lines, they are cut in the same ratio. 2. If, on two bases, there be placed the same number of triangles, similar each to each, and similarly situated, the polygons formed by joining the vertices of the triangles will be similar. 3. A straight line, drawn from the vertex of a triangle to meet the base, divides a parallel to the base in the same ratio as the base. 4. If, from two angular points of a triangle, straight lines be drawn to bisect the opposite sides, they will divide each other into segments, having the ratio of two to one. 5. From a given point, either within or without a given angle, to draw a straight line cutting the two lines which contain the angle, so that the distances of the two intersections from the given point shall be to one another in a given ratio. 6. To describe a right-angled triangle that shall have a given hypothenuse, and its three sides continually proportional. EDINBURGH, PRINTED BY JOHN STARK. THE END. |