every straight line which joins a point of one of these lines to a point of the other. See (526). 13. The projection of a straight line on a plane is a straight line which is either parallel to the first straight line or meets it in the point where it cuts the plane, according as the first line is parallel to the plane or not. 14. From a point P, PA and PB are drawn perpendicular to two planes which intersect in CD, meeting them in A and B; from A, AE is drawn perpendicular to CD: prove that BE is also perpendicular to CD. 15. Two planes which are not parallel are cut by two parallel planes: prove that the intersections of the first two with the last two contain equal angles. See (519), (525). 16. If two parallel planes be cut by three other planes which have a point, but no line common to all three, and no two of which are parallel, the triangles formed by the intersections of the parallel planes with the other three planes are similar to each other. See (519), (525). 17. The projections of parallel straight lines on any plane are themselves parallel. See (509), (525), (519). 18. Two straight lines which intersect are inclined to each other at an angle equal to two-thirds of a right angle, and to a given plane, each, at an angle equal to half a right angle. Prove that their projections on this plane are at right angles to each other. 19. In any triedral angle, the planes bisecting the three diedral angles, all intersect in the same straight line. See (549). 20. In any triedral angle, the planes which bisect the three face-angles, and are perpendicular to these faces respectively, all intersect in the same straight line. From the vertex S, take equal distances SA, SB, SC, on the three edges; the intersections of the three 1 bisecting planes with the plane of ABC are 1 bisecting lines of the sides of ABC, and bave a common intersection (168); . . etc. 21. In any triedral angle, the three planes which pass through the edges, perpendicular to the opposite faces respectively, all intersect in the same straight line. See (516). 22. ABCD is a face and AE a diagonal of a cube, BG is drawn perpendicular to AE, and DG is joined: prove that DG is perpendicular to AE. 23. If two face-angles of a triedral angle are equal, the diedral angles opposite them are also equal. 24. An isosceles triedral angle and its symmetrical triedral angle are equal. 25. If BAC, CAD, DAB be the three face-angles of a triedral angle, prove that the angle between AD and the straight line bisecting the angle BAC is less than half the sum of the angles BAD, CAD. See (565). 26. A triedral angle is contained by the three face-angles BOC, COA, AOB; if BOC, COA are together equal to two right angles, prove that CO is perpendicular to the line which bisects the angle AOB. Loci. 27. Find the locus of points which are equally distant from three given points not in the same straight line. 28. Find the locus of points which are equally distant from two given intersecting straight lines. 29. Find the locus of points which are equally distant from two given parallel planes; or whose distances from the parallel planes are in a given ratio. 30. Find the locus of points which are equally distant from three given planes. See (549). 31. Find the locus of points which are equally distant from the three edges of a triedral angle. 32. Find the locus of points which are equally distant from the three faces of a triedral angle. 33. Find the locus of points which are equally distant from two given planes, and at the same time equally distant from two given points. 34. Find the locus of a point such that the sum of its distances from two given planes is equal to a given straight line. 35. Find the locus of a point such that the sum of its distances from three given planes is equal to a given straight line. PROBLEMS. 36. Pass a plane perpendicular to a given straight line through a given point not in that line. 37. Pass a plane through a given straight line, perpendicular to a given plane. See (513). 38. Pass a plane through a given point parallel to a given plane. See (522). 39. Find the point in a given straight line which is equally distant from two given points not in the same plane with the given line. 40. Draw a straight line through a given point in space, so that it shall cut two given straight lines not in the same plane. 41. Draw a straight line through a given point in a given plane, so that it shall be perpendicular to a given line in space. 42. Two given straight lines do not intersect and are not parallel: find a plane on which their projections will be parallel. 43. Divide a straight line similarly to a given divided straight line lying in a different plane. Let ACDB be the divided line, NMLF the other line; draw EHKG II to AB, and CH, DK, BG || to AE; also KL, HM, EN || to GF: :. (525), (526). 44. Given three straight lines meeting at a point: draw through the given point a straight line equally inclined to the three. BOOK VII. POLYEDRONS. DEFINITIONS. 568. A polyedron is a solid bounded by planes. The portions of the bounding planes, limited by their mutual intersections, are the faces of the polyedron; the intersections of the faces are the edges, and the intersections of the edges are the vertices, of the polyedron. A diagonal of a polyedron is a straight line joining two vertices not in the same face. 569. A polyedron of four faces is called a tetraedron; one of six faces, a hexaedron; one of eight faces, an octaedron; one of twelve, a dodecaedron; one of twenty, an icosaedron. 570. A polyedron is convex when the section made by any plane intersecting it is a convex polygon. Only convex polyedrons are treated of in this work. 571. The volume of a solid is its numerical measure (229), referred to some other solid called the unit of vol ume. 572. Two solids are equivalent when they have equal volumes. 573. Two polygons are said to be parallel when their sides are respectively parallel. PRISMS AND PARALLELOPIPEDS. 574. A prism is a polyedron two of whose faces are equal and parallel polygons, and the other faces are parallelograms. The equal and parallel polygons are called the bases of the prism; the parallelograms are the lateral faces; the lateral faces taken together form the lateral or convex surface; and the intersections of the lateral faces are the lateral edges. The lateral edges are parallel and equal (511) and (520). The area of the lateral surface is called the lateral area. 575. The altitude of a prism is the perpendicular distance between its bases. 576. A prism is said to be triangular, quadrangular, hexagonal, etc., according as its bases are triangles, quadrilaterals, hexagons, etc. 577. A right prism is one whose lateral edges are perpendicular to its bases. A regular prism is a right prism whose bases are regular polygons: therefore its lateral faces are equal rectangles. An oblique prism is one whose lateral edges are not perpendicular to its bases. 578. A right section of a prism is a section by a plane perpendicular to its lateral edges. 579. A truncated prism is the part of a prism included between the base and an oblique section made by a plane cutting all the lateral edges. 580. A parallelopiped is a prism whose bases are parallelograms: hence all the faces are parallelograms. 581. A right parallelopiped is one whose lateral edges are perpendicular to the bases: hence the lateral faces are rectangles. |