6. Show that the locus of points whose distances from two given intersecting planes are in a given ratio is a plane. 7. Show that the locus of points whose distances from three intersecting planes are (a) equal, or (b) in a given ratio, is a straight line. 8. If every face of a parallelopiped is a rectangle prove that its four diagonals are all equal. 9. Find by construction the angle between any two diagonals of a cube. 10. Find the locus of a point which moves so that at any instant it is equidistant from all points on the circumference of a given circle. 11. Each edge of a tetrahedron is equal to the opposite edge. Show that all four faces are congruent. 12. Prove that the four middle points of two pairs of opposite edges of a tetrahedron are coplanar. 13. The six planes each passing through one edge of a tetrahedron and bisecting the opposite edge meet in a point. 14. ABCD is a regular tetrahedron and AP is drawn perpendicular to the plane BCD. Prove that (i) AP is equally inclined to AB, AC, AD, (ii) P is circumcentre of ▲ ABC, and (iii) CĎ is perpendicular to the plane APB. 15. A tetrahedron ABCD has each edge 2 inches long. Calculate the length of the perpendicular AP from A to the opposite face. 16. In the previous question calculate the inclination of the edge AB to the face BCD, and the dihedral angle between the two faces ACD, BCD. [Use Mathematical Tables.] 17. In Question 15, calculate the radii of the circumscribed and inscribed spheres. 18. In Question 15, calculate the distance between the middle points of opposite edges of the tetrahedron. 19. In the tetrahedron ABCD, AB = CD = 4", AC = BD = 5′′, AD = BC = 6". Calculate the edges of the circumscribing parallelopiped, and the lengths of the three lines joining middle points of opposite sides of the tetrahedron. 20. In the tetrahedron ABCD, AB=AC=AD=2", BC=CD=DB=3". Calculate the radius of the circumscribing sphere. 21. Prove that a sphere can be drawn through any two circles, not in the same plane, which cut one another in two points. If the planes of the two circles are at right angles, if their radii are a, B, and the length of their common chord 2c, show that the square of the radius of the sphere is a2 + 82 – c2. 22. Three equal spheres each 1 inch in diameter rest in contact with each other in a spherical bowl whose diameter is 6 inches. Find the distance of the plane passing through the centres of the spheres from the centre of the bowl. 23. Through a fixed point 0 on a given sphere a straight line is drawn cutting the sphere at P, and on OP a point Q is taken such that the rectangle OP, OQ is equal to a given square. Show that lies on a fixed plane whose distance from O is a third proportional to the diameter of the sphere and the side of the given square. 24. Prove that every section of a tetrahedron made by a plane parallel to two opposite edges is a parallelogram. When is this parallelogram (a) a rectangle, (b) a square? 25. A plane section of a tetrahedron is a parallelogram. Prove that the plane of section is parallel to two opposite edges of the tetrahedron. 26. The shortest distance between the diagonal of a cube and any edge which does not intersect this diagonal is the straight line joining their middle points. 27. Show how to cut a cube by a plane so that the lines of section may form a regular hexagon. 28. In a tetrahedron the line joining the middle points of two opposite edges is perpendicular to both. Prove that of the other four edges any one is equal in length to the opposite one. 29. A parallelopiped is circumscribed to a tetrahedron (see § 440). Prove that- (i) If each edge of the tetrahedron is equal to the opposite edge the parallelopiped is rectangular. (ii) If the tetrahedron is regular the parallelopiped is a cube. (iii) If the tetrahedron is orthocentric the parallelopiped has all its edges equal. 30. Can the orthocentre of an orthocentric tetrahedron lie (i) at a vertex, (ii) on an edge but not at a vertex, (iii) on a face but not on an edge? 31. Each edge of a tetrahedron is equal to the opposite edge. Prove that the sum of the squares on the edges is equal to four times the sum of the squares on the three lines joining the middle points of opposite edges. 32. In the tetrahedron ABCD, given that AB2+ CD2 AC2 + BD2 = AD2 + BC2, = prove that each edge is perpendicular to the opposite edge. LIST OF PROPOSITIONS. THE chief object of this list of Propositions is to serve for an Index to the reference symbols used throughout the book. The list will also be found useful as a summary of the course of Theoretical Geometry. Angles. CHAPTER II.-ANGLES AND PARALLELS. A. 1.-The two angles which one straight line makes with another straight line on one side of it are together equal to two right angles PAGE (Euc. I. 13) 83 A.1c.-If at a point in a straight line, two other straight lines on opposite sides of it make the adjacent angles together equal to two right angles, these two straight lines shall be in the same straight line (Euc. I. 14) A.2.-Each of the angles formed by two intersecting straight lines is equal to the vertically opposite angle Parallels. 84 (Euc. I. 15) 86 P.1. If a straight line meeting two other straight lines in the same plane makes two alternate angles equal to one another, these two straight lines shall be parallel (Euc. I. 27) P.2.-If a straight line, meeting two other straight lines, makes an exterior angle equal to the interior and opposite angle on the same side of the line; or if it makes the two interior angles on the same side together equal to two right angles; then the two straight lines shall be parallel (Euc. I. 28) PLAYFAIR'S AXIOM.-It is not possible that there should be two straight lines drawn through the same point and parallel to the same line P.1c. If a straight line meets two parallel straight lines, it shall make the alternate angles equal to one another (Euc. I. 29) P.2c.—If a straight line meets two parallel straight lines, it shall make each exterior angle equal to the interior and opposite angle on the same side of the line, and each pair of interior angles on the same side of the line together equal to two right angles (Euc. I. 29) P.3.-If two straight lines are parallel to the same straight line they are parallel to one another (Euc. I. 30) P.4.-If the two arms of one angle are respectively parallel to the two arms of another, then the angles are either equal or supplementary . P.5. [See Chapter IV.] Triangles. CHAPTER III.-TRIANGLES. PAGE T.1.—If any side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior and opposite angles: also, the three angles of a triangle are together equal to two right angles (Euc. I. 32) 105 RT.1.t-In a right-angled triangle the two acute angles are together equal to one right angle. Polygons. 106 Pn.1. If all the sides of a convex polygon be produced in the same direction round the polygon, the exterior angles so formed are together equal to four right angles; and in any convex polygon the sum of the interior angles, together with four right angles, is equal to twice as many right angles as the figure has sides (Euc. I. 32, Cor.) 110 Congruence. C.1.—If two triangles have two sides and the included anglè in the one respectively equal to two sides and the included angle in the other, then shall these triangles be congruent. (Euc. I. 4) 114 C.2. If two triangles have two angles and one side in the one respectively equal to two angles and the corresponding side in the other, the two triangles shall be congruent. Triangles. (Euc. I. 26) 116 T.2. If two sides of a triangle are equal, then shall the angles opposite to these sides be equal (Euc. I. 5) 118 T. 2c.-If two angles of a triangle are equal, then the sides opposite to these angles are also equal Congruence. (Euc. I. 6) 119 C.3.-If the three sides of one triangle are respectively equal to the three sides of another triangle, the two triangles shall be congruent (Euc. I. 8) 120 C.4.-If the hypotenuse and one side of one right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the two triangles shall be congruent. 122 C.5.-If two sides of one triangle are respectively equal to two sides of another, and if the angles opposite to one pair of equal sides are equal, then shall the angles opposite to the other pair of equal sides be either equal or supplementary. (Cf. Euc. VI. 7) 125 For the remaining Theorems on Right-angled Triangles see Chapters IX, XXI. For the remaining Theorems on Polygons see Chapters XVIII, XX. PAGE Triangles. T.3.-If one side of a triangle is greater than another, then the angle opposite to the first side shall be greater than the angle opposite to the second (Euc. I. 18) T.3c.-If one angle of a triangle is greater than another, then the side opposite to the first angle shall be greater than the side opposite to the second (Euc. I. 19) T.4. Any two sides of a triangle are together greater than the third (Euc. I. 20) T.5. The shortest line which can be drawn from a given point to a given line is the line which is perpendicular to the given line; and of any other two lines drawn from the given point to the given line on one side of the perpendicular, that which is nearer to the perpendicular is less than that which is more remote T.6.-If the two sides of one triangle are respectively equal to the two sides of another, and if the vertical angle of the first triangle is greater than the vertical angle of the second triangle; then the base of the first triangle shall be greater than the base of the second (Euc. I. 24) T.6c. If the two sides of one triangle are respectively equal to the two sides of another, and if the base of the first triangle is greater than the base of the second triangle; then the vertical angle of the first triangle shall be greater than the vertical angle of the second (Euc. I. 25) T.7.-[See Chapter IV.] T.8, 9, 10.-[See Chapter XVI.] T.11, 11c, 12, 13.-[See Chapter XIX.] T. 14, 15, 16.—[See Chap. XXII.] T. 17, 18.-[See Chap. XXV.] 123 123 124 128 129 130 CHAPTER IV.-PARALLELOGRAMS. Parallelograms. Pm.1.-In any parallelogram, opposite sides are equal, and opposite angles are equal; also each diagonal divides the parallelogram into two congruent triangles (Euc. I. 34) 136 Pm.1c.-(i) If the opposite sides of a quadrilateral are equal, it is a parallelogram; also, (ii) if the opposite angles of a quadrilateral are equal it is a parallelogram 137 Pm.2.-If in a quadrilateral two opposite sides are equal and parallel, then the quadrilateral is a parallelogram, and the other two sides are also equal and parallel (Euc. I. 33) 138 |