161. Given three straight lines in space.

from the first to the second parallel to the third.

Draw a straight line

162. Given two straight lines not in the same plane. Find a point in one at a given perpendicular distance from the other.

163. Through a given point draw a straight line to meet a given straight line and the circumference of a given circle not in the same plane with the given line.

BOOK VII

PROBLEMS OF DEMONSTRATION

164. A triangular pyramid is cut by a plane parallel to the base, and a plane is passed through each vertex of the base and the points where the cutting plane meets the two opposite lateral edges. Determine the locus of the point of intersection of the three planes thus passed.

165. At any point in the base of a regular pyramid a perpendicular to the base is erected, intersecting the lateral faces of the pyramid, or these faces produced. Prove that the sum of the perpendicular distances from the points of intersection to the base is constant.

166. The perpendicular from the centre of gravity of a tetraedron (§ 749) to any plane is one-fourth the sum of the four perpendiculars from the vertices of the tetraedron to the same plane.

167. If the edges of a hexaedron meet four by four in three points, the four diagonals of the hexaedron meet in a point.

168. Prove that straight lines through the middle points of the sides of any face of a tetraedron each parallel to the straight line connecting a fixed point D with the middle point of the opposite edge, meet in a point E such that DE passes through and is bisected by the centre of gravity of the tetraedron.

169. The sum of the perpendiculars drawn to the faces of a regular tetraedron from any point within is equal to the altitude of the tetraedron.

170. A regular octaedron is cut by a plane parallel to one of its faces; prove that the perimeter of the section is constant.

171. In a tetraedron the sum of two opposite edges is equal to the sum of two other opposite edges. Prove that the sum of the diedral angles whose edges are the first pair of lines is equal to the sum of the diedral angles whose edges are the other pair of lines.

172. C' and D' are the feet of the perpendiculars drawn from any point to the faces opposite the vertices C, D of a tetraedron ABCD. Prove that AC"—BC"=AD2—BD".

173. If the opposite edges of a tetraedron are perpendicular to each other, the perpendiculars drawn from the vertices to the opposite faces meet in a point.

174. If a tetraedron is cut by a plane which passes through the middle points of two opposite edges, the section is divided into two equivalent triangles by the straight line joining these points.

175. From the middle point of one of the edges of a regular tetraedron a fly descends by crawling around the tetraedron, and reaches the base at the point where this edge meets the base. Find at what points the fly must cross the other edges if its path is everywhere equally inclined to the plane of the base.

176. The plane bisecting a diedral angle of a tetraedron divides the opposite edge into segments which are proportional to the faces which include the diedral angle.

177. Straight lines are drawn from the vertices A, B, C, D of a tetraedron through a point P, to meet the opposite faces in A', B', C', D'. Prove that

Ρ.Α' PB' PC' PD'
+ + + =1
AA' BB' CC' DD'

178. If a is the edge of a regular tetraedron, its volume is

a3

179. If a is the edge of a regular octaedron, its volume is

12 V2.

a3

3

180. The lateral surface of a pyramid is greater than its base. 181, The volume of a triangular prism is equal to the area of a lateral face multiplied by one-half its perpendicular distance from any point in the opposite lateral edge.

182. The volume of a regular prism is equal to the product of its lateral area by one-half the apothem of its base.

183. The three lateral faces of a tetraedron are perpendicular to each other. If a triangle drawn in the base is projected on each of the three lateral faces, prove that the sum of the pyramids having these projections as bases and a common vertex anywhere in the base of the given tetraedron is equivalent to the pyramid having the given triangle for its base and its vertex at the vertex of the given tetraedron.

184. Extend the last exercise to the case where the common vertex is at any point in the plane of the base by regarding the volume of a pyramid as negative if the altitude is in the opposite direction from that in which it was measured for the pyramid on the same base in the last exercise.

185. Defs.—If ABCD is a rectangle, and EF a straight line parallel to AB, and not in the plane of the rectangle, the solid bounded by the rectangle ABCD, the trapezoids ABFE, CDEF, and the triangles ADE. BCF is a wedge.

The rectangle is called the back of the wedge; the trapezoids, its faces; the triangles, its ends; the line EF, its edge; AB is the length of the back and AD its breadth; the perpendicular from any point of EF upon the back is the altitude of the wedge.

186. If h is the altitude, prove that the volume of the above wedge is

hx AD×(2AB+EF).

187. Defs. If ABCD and EFGH are two rectangles lying in parallel planes, AB and BC being parallel to EF and FG, respectively, the solid bounded by these two rectangles and the trapezoids ABFE, BCGF, CDHG, DAEH, is called a rectangular prismoid. The rectangles are called the bases of the prismoid and the perpendicular distance between them the altitude.

188. Prove that the volume of a rectangular prismoid is equal to the product of the sum of its bases, plus four times a section equidistant from the bases, multiplied by one-sixth the altitude.

PROBLEMS OF CONSTRUCTION

189. Having given the four perpendiculars from the vertices of a tetraedron to the opposite faces, and the distance of a point in space from three of the faces, find its distance from the fourth face.

190. Through a given straight line in one of the faces of a tetraedron pass a plane which shall cut off from the tetraedron another tetraedron which is to the first in a given ratio.

191. Find two straight lines whose ratio shall be the ratio of the volumes of two given cubes.

192. Find a point within a given tetraedron, such that the four pyramids having this point for vertex, and the faces of the tetraedron for bases, shall be equivalent.

PROBLEMS FOR COMPUTATION

193. (1.) Find the lateral area, total area, and volume of a regular triangular prism the perimeter of whose base is 16.413 in. and whose altitude is 14.718 in.

(2.) Find the lateral area, total area, and volume of a regular hexagonal pyramid each side of whose base is 8.84 in. and whose altitude is 4.92 in.

(3.) The area of the base of a pyramid is 13 sq. m.; its altitude is 4 m. Find the area of a section parallel to the base and distant 11⁄2 m. from it. Also find the volume of the pyramid cut off by this plane.

(4.) Find the volume of a frustum of a pyramid whose base is a regular octagon having each side equal to 4 in., and whose altitude is 9 in., made by a plane 5 in. from the vertex.

(5.) The diagonal of a cube is 24.16 cm. Find its surface and volume. (6.) The volume of a polyedron is 984.62 cu. ft. Find the volume of a similar polyedron whose edges are nine times the edges of the first polyedron.

(7.) The volume of a given tetraedron is 6.86 cu. m. Find the volume of the tetraedron whose vertices are a vertex of the given tetraedron and the intersections of the medians of the faces including that

vertex.

(8.) Find the surface and volume of a regular tetraedron whose edge is I.

(9.) Find the surface and volume of a regular octaedron whose edge is 16.247 mm.

(10.) Find the ratio of the volumes of a cube and a regular tetraedron whose edges are equal.

(11.) Find the ratio of the volumes of a regular octaedron and a regular tetraedron whose edges are equal.

(12.) Find the number of cubic feet of water that will be contained by a trench in the shape of a wedge the length of whose back is 20 m., whose breadth is 3 m., whose edge is 16 m., and whose depth is 24 m. How many pounds of water will the trench hold, each cubic foot of water weighing 62 lbs.? How many metric tons?

(13.) An embankment is in the form of a rectangular prismoid. The length and breadth of its base are 246 ft. and 8 ft.; the length and breadth of its top are 239 ft. and 3 ft. Its height is 4 ft. Find the number of cubic yards of earth it contains.

BOOK VIII

PROBLEMS OF DEMONSTRATION

194. If two circles in space are such that their centres are the projections of the same point on their planes, and the tangents to the circles drawn from a point in the intersection of their planes are equal, the two circles are on the same sphere.

195. If through a fixed point within or without a sphere three straight lines are drawn at right angles to each other so as to intersect the surface of the sphere, the sum of the squares of the three chords thus formed is constant. Also the sum of the squares of the six segments of these chords is constant.

196. If three radii of any sphere perpendicular to each other are projected upon any plane, the sum of the squares of the three projections is equal to twice the square of the radius of the sphere.

197. If from a point without a sphere any number of straight lines be drawn to touch the sphere, the points of contact will all be in one plane.

198. A sphere can be inscribed in or circumscribed about any regular polyedron.