drawn, each passing through two of these lines, such that the perpendiculars, from any point in the line of intersection of the given planes, upon any one of the four planes, shall be equal to the given line. 1859. VI. 31. Shew that, on a given straight line, there may be described as many polygons of different magnitudes, similar to a given polygon, as there are sides of different lengths in the polygon. XI. 20. Three straight lines, not in the same plane, intersect in a point, and through their point of intersection another straight line is drawn within the solid angle formed by them : prove that the angles, which this straight line makes with the first three, are together less than the sum, but greater than half the sum of the angles which the first three make with each other. 1860. VI. A. If the two sides, containing the angle, through which the bisecting line is drawn, be equal, interpret the result of the proposition. Prove from this proposition and the preceding, that the straight lines, bisecting one angle of a triangle internally, and the other two externally, pass through the same point. XI. 17. If three straight lines, which do not all lie in one plane, be cut in the same ratio by three planes, two of which are parallel, shew that the third will be parallel to the other two, if its intersections with the three straight lines are not all in one straight line. 1861. vi. 6. From the angular points of a parallelogram ABCD, perpendiculars are drawn on the diagonals, meeting them in E, F, G, H respectively ; prove that EFGH is a parallelo gram similar to ABCD. 1861. XI. 12. Shew that the shortest distance between two opposite edges of a regular tetrahedron is equal to half the diagonal of the square, de scribed on an edge. 1862. vi. 1. Lines are drawn from two of the angular points of a triangle, to divide the opposite sides in a given ratio ; prove that the line, joining the third angular point with the point of intersection of these two lines, either bisects the opposite side, or divides it in a ratio which is the duplicate of the given ratio. XI. 21. If four points be so situated that the distance between each pair is equal to the distance between the other pair, prove that the angles subtended at any one of these points by each pair of the others, are together equal to two right angles. 1863. VI. 4. The internal angles at the base of a triangle, and the external angle at the vertex, are bisected by straight lines ; prove that the three points, in which these straight lines meet the opposite sides respectively, lie on one straight line. XI. 17. If each edge of a tetrahedron be equal to the opposite edge, the straight line, joining the middle points of any two opposite edges, shall be at right angles to each of those edges. 1864. VI. 23. If one parallelogram have to another parallelo gram the ratio, which is compounded of the ratios of their sides, one parallelogram shall be equiangular. XI. 12. On a given equilateral triangle describe a regular tetrahedron. 1865. vi. 19. The opposite sides, BA, CD of a quadilateral ABCD, which can be inscribed in a circle, meet, when produced, in E; F is the point of intersection of the diagonals, and EF meets AD in G: prove that the rectangle EA, AB is the rectangle ED, DC as AG is to GD. XI. 16. In the triangular pyramid ABCD, AB is at right angles to CD, and AC to BD: prove that AD is at right angles to BC. 1866. VI. 4. ABC is an isosceles triangle ; AE is the perpen dicular from A on the base BC; D is any DE is double of the ratio of AF to FB. circles are to one another as the squares on their diameters. 1867. VI. A. Each acute angle of a right-angled triangle and its corresponding exterior angle are bisected by straight lines meeting the opposite sides ; prove that the rectangle, contained by the portions of those sides intercepted between the bisecting lines is four times the square on the hypotenuse. XI. 21. Two pyramids are described, the one standing on a square as a base, the other on a regular octagon, the vertex of each being equally distant from the angular points of its base ; if this distance be the same for each pyramid, and the perimeters of the bases be equal, prove that the plane angles, containing one solid angle at the vertex of the former, are together greater than the plane angles, containing the sclid angle at the vertex of the latter. 1868. VI. 2. Without assuming any subsequent proposition, prove that the equiangular triangles in either of the figures of this proposition, are to each other in the duplicate ratio of the sides oppo site to the equal angles. 1868. XI. 11. Of the least angles, which a given line in one plane makes with any line in another plane, the greatest for different positions of the given line is that which measures the inclina tion of the two planes. 1869. XI. 20. If O be a point, within a tetrahedron ABCD, prove that the three angles of the solid angle, subtended by BCD at 0, are together greater than the three angles of the solid angle at A. 1870. vi. 15. Two straight lines are given in position, and a third straight line is drawn so as to cut off a triangle equal to a given triangle; through the middle point of this third side is drawn a straight line in a given direction, terminated by the two given straight lines : prove that the rectangle under the segments of the intercepted part is constant. XI. 7. In a tetrahedron each edge is perpendicular to the direction of the opposite edge ; prove that the straight line joining the centre of the sphere, circumscribing the tetrahedron, to the middle point of any edge, is equal and parallel to the straight line joining the centre of perpendiculars to the middle point of the opposite edge. 1871. vi. 2. ABC is a triangle, and lines AO, BO, CO cut the opposite sides in D, E, F; if EF cut BC in G, prove that BD is to DC as BG is to GC. xi. 11. The perpendiculars from the angular points of a tetrahedron on the opposite faces meet in a point : prove that the necessary and sufficient condition for this is that the sums of the squares of pairs of opposite edges be equal. 1872. VI. 2. Draw through a point a straight line, so that the part of it intercepted between a given straight line and a given circle may be divided at the given point in a given ratio. Between what limits must the ratio lie in order that a solution may be possible? XI. 20. If the opposite edges of a tetrahedron be equal two and two, prove that the faces are acuteangled triangles. Prove also that a tetrahedron can be formed of any four equal and similar acute-angled triangles. PRINTED BY T. AND A. CONSTABLE, PRINTERS TO AER MAJESTY, AT THE EDINBURGH UNIVERSITY PRESS. |