PROPOSITION C. THEOREM. If from the vertical angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle. Let ABC be a triangle, and let AD be the perpendicular from the angle A to the base BC: the rectangle BA, AC shall be equal to the rectangle contained by AD and the diameter of the circle described about the triangle. Describe the circle ACB about the triangle; [IV. 5. draw the diameter AE, and join EC. Then, because the right angle BDA is equal to the angle ECA in a semicircle; [III. 31. and the angle ABD is equal to the angle AEC, for they are in the same segment of the circle; B [III. 21. therefore the triangle ABD is equiangular to the triangle AEC. Therefore BA is to AD as EA is to AC; [VI: 4. therefore the rectangle BA, AC is equal to the rectangle EA, AD. [VI. 16. Wherefore, if from the vertical angle &c. Q.E.D. PROPOSITION D. THEOREM. The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to both the rectangles contained by its opposite sides. Let ABCD be any quadrilateral figure inscribed in a circle, and join AC, BD: the rectangle contained by AC, BD shall be equal to the two rectangles contained by AB, CD and by AD, BC. Make the angle ABE equal to the angle DBC; [I. 23. add to each of these And the angle BDA is B Therefore AD is to DB as EC is to CB ; [VI. 4. therefore the rectangle AD, CB is equal to the rectangle DB, EC. [VI. 16. Again, because the angle ABE is equal to the angle DBC, [Construction. and the angle BAE is equal to the angle BDC, for they are in the same segment of the circle; [III. 21. therefore the triangle ABE is equiangular to the triangle DBC. Therefore BA is to AE as BD is to DC; [VI. 4. therefore the rectangle BA, DC is equal to the rectangle AE, BD. [VI. 16. But the rectangle, AD, CB has been shewn equal to the rectangle DB, EC; therefore the rectangles AD, CB and BA, DC are together equal to the rectangles BD, EC and BD, AE; that is, to the rectangle BD, AC. [II. 1. Wherefore, the rectangle contained &c. Q.E.D. BOOK XI. DEFINITIONS. 1. A SOLID is that which has length, breadth, and thickness. 2. That which bounds a solid is a superficies. 3. A straight line is perpendicular, or at right angles, to a plane, when it makes right angles with every straight line meeting it in that plane. 4. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicular to the common section of the two planes, are perpendicular to the other plane. 5. The inclination of a straight line to a plane is the acute angle contained by that straight line, and another drawn from the point at which the first line meets the plane to the point at which a perpendicular to the plane drawn from any point of the first line above the plane, meets the same plane. 6. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the same point of their common section at right angles to it, one in one plane, and the other in the other plane. 7. Two planes are said to have the same or a like inclination to one another, which two other planes have, when the said angles of inclination are equal to one another. 8. Parallel planes are such as do not meet one another though produced. 9. A solid angle is that which is made by more than two plane angles, which are not in the same plane, meeting at one point. 10. Equal and similar solid figures are such as are contained by similar planes equal in number and magnitude. [See the Notes.] 11. Similar solid figures are such as have all their solid angles equal, each to each, and are contained by the same number of similar planes. 12. A pyramid is a solid figure contained by planes which are constructed between one plane and one point above it at which they meet. 13. A prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to one another; and the others are parallelograms. 14. A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains fixed. 15. The axis of a sphere is the fixed straight line about which the semicircle revolves. 16. The centre of a sphere is the same with that of the semicircle. 17. The diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the superficies of the sphere. 18. A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed. If the fixed side be equal to the other side containing the right angle, the cone is called a right-angled cone; if it be less than the other side, an obtuse-angled cone; and if greater, an acute-angled cone. 19. The axis of a cone is the fixed straight line about which the triangle revolves. 20. The base of a cone is the circle described by that side containing the right angle which revolves. 21. A cylinder is a solid figure described by the revolution of a right-angled parallelogram about one of its sides which remains fixed. 22. The axis of a cylinder is the fixed straight line about which the parallelogram revolves. 23. The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram. 24. Similar cones and cylinders are those which bave their axes and the diameters of their bases proportionals. 25. A cube is a solid figure contained by six equal squares. 26. A tetrahedron is a solid figure contained by four equal and equilateral triangles. 27. An octahedron is a solid figure contained by eight equal and equilateral triangles. 28. A dodecahedron is a solid figure contained by twelve equal pentagons which are equilateral and equiangular. 29. An icosahedron is a solid figure contained by twenty equal and equilateral triangles. A. A parallelepiped is a solid figure contained by six quadrilateral figures, of which every opposite two are parallel. PROPOSITION 1. THEOREM. One part of a straight line cannot be in a plane, and another part without it. It it be possible, let AB, part of the straight line ABC, be in a plane, and the part BC without it. |