jected line lie, in order that its projection upon a plane shall be a straight line (a) if it is straight? (b) if it is not straight? If a line when projected upon each of two planes makes straight lines on both, what must be true of it? (Two cases.) Give the determining lines that must be drawn in order to construct a plane (1) through a point (a) parallel to a given plane, (b) parallel to two non-coplanar lines, (c) perpendicular to a line, (d) perpendicular to a plane; (2) through a line (a) parallel to a non-coplanar line, (b) perpendicular to a plane. State two methods that have been used to prove two lines perpendicular. What methods have been found to prove angles equal? unequal? sects equal? unequal? sects proportional? Upon what plane geometry locus theorems can the following be made to depend: locus of points equidistant from two points? three points? two lines? two planes? State all the propositions that prove that there can be a line or a plane that fulfills certain conditions; all propositions that prove that there can be but one line or one plane that fulfills certain conditions.

GENERAL EXERCISES

82. If two perpendiculars to a plane from two points on the same side of it are equal, the line joining those points is parallel to the plane. If three or more such perpendiculars are equal, the points determine a plane parallel to the given plane.

83. If two planes are drawn perpendicular respectively to two non-coplanar lines, their intersection is perpendicular to any plane parallel to the given lines.

84. Find the locus of points in a given plane, equidistant from two given planes.

NOTE. Two points equidistant from a plane, on the same perpendicular to that plane, are called symmetric with regard to that plane.

85. Prove that two points symmetric to a plane are equidistant from any point in the plane.

86. If A and B are on the same side of plane M, find the shortest path between A and B that includes one point of M. (Use the point symmetric to either A or B.) This is simply finding the shortest way to go from A to M and back to B. It is the path that a ray of light travels when it meets a mirror and is reflected; it is the path followed by a billiard ball striking a cushion and bounding back, or

by a tennis ball bounding from the ground, unless, in either of these cases, the ball is affected by some motion, such as whirling, other than the mere rebound.

87. If two balls are in positions A and B on a billiard table, show how the point on a cushion could be determined so that if A strikes the cushion at that point, it will rebound to B. Show how a point could be determined so that A would strike two cushions and rebound to B; three cushions; all four cushions. Note that in the last cases symmetry is being used in regard to two or more planes.

88. Find the shortest path in two intersecting planes between a point in one of the planes, and a point in the other plane.

89. Find the shortest path in the surface of a box from one corner to the opposite corner.

90. If two lines in one of two intersecting planes make equal angles with the intersection, they make equal angles with the other plane, and conversely.

91. Perpendiculars from a point to a set of parallel lines are coplanar.

92. The arms of an angle are equally inclined to any plane through its bisector.

93. The arms of an angle are equally inclined to any plane through a coplanar line perpendicular to its bisector.

94. If a line is perpendicular to one of two intersecting planes, the projection upon the other plane is perpendicular to the intersection.

95. If any number of planes perpendicular to the same plane have a common point, they intersect in a common line.

96. If a plane and a line are parallel, a plane perpendicular to the line is also perpendicular to the plane. Is the converse also true?

97. If three or more lines are drawn from points on one of two non-coplanar lines pèrpendicular to the other line, the sects cut off on the two non-coplanar lines by these perpendiculars are propor

tional.

98. If a common perpendicular is drawn to two lines that are either parallel or non-coplanar, the plane perpendicular to that common perpendicular at its midpoint bisects every transversal of the two lines.

99. If two congruent rectangles are placed so that they have an edge in common but are not coplanar, their common edge is perpendicular to two of the planes determined by other edges, and is parallel to a third plane. State the general case.

100. When will lines perpendicular to two intersecting planes meet? Prove it.

101. Planes perpendicular to intersecting lines meet. What can be told about their intersection?

102. Given three lines, no two of which are coplanar: (a) pass a plane through one of them so as not to be parallel to either of the others; (b) draw a line through all three of them. Can a line be drawn through one of them parallel to the other two? a plane?

103. Given a line lying within and parallel to the edge of a dihedral angle, how could a plane be passed through this line, intersecting the faces of the angle, so that the parallels so formed would be equidistant from the given line?

104. Given a line outside a dihedral angle parallel to its edge, how could a plane be passed through this line meeting the faces of the dihedral angle so that the distance apart of the parallel intersections shall equal the distance of one of them from the given line.

105. If a circle is inscribed in a triangle, the lines from any point in the perpendicular erected to the plane of the circle at the center to the points of contact are perpendicular to the sides of the triangle.

106. The lines from any point in the perpendicular to a plane at the incenter of a given triangle, perpendicular to the sides of the triangle, meet them at the points where they are tangent to the inscribed circle.

107. If in each of two intersecting planes two lines are drawn parallel to the intersection, and such that all four lines are the same distance from the intersection, those lines determine two pairs of parallel planes.

108. If four lines determine more than four planes, the lines are either all concurrent or all parallel.

109. If two pairs of parallel planes intersect, the other two planes determined by their intersections meet in a line parallel to the given planes and to their intersections, and equidistant from each opposite pair of intersections.

110. If one pair of parallel planes is perpendicular to a second pair of parallel planes, the distance between one pair of opposite intersections equals the distance between the other pair of opposite intersections.

111. If three planes meet in parallel lines, and two of them make equal angles with the third, then the bisector of the dihedral angle between those two planes is perpendicular to the third plane.

112. If two pairs of parallel planes intersect, and the consecutive pairs of their intersections are the same distance apart, the diagonal planes of the figure formed are perpendicular to each other.

113. If two parallel planes are crossed by two non-parallel planes that make equal angles with one of them,

(a) those planes make equal angles with the other of the parallel planes;

(b) the opposite intersections are equidistant;

(c) the non-parallel planes meet in a line parallel to the given intersections, and equidistant from those in either of the parallel planes;

(d) the given intersections in one of the non-parallel planes are the same distance apart as the intersections in the other of those planes. (e) State and prove the converse of (d).

114. If two triangles (as ABC and XYZ) in different planes are so placed that each corresponding pair of sides intersect (as, AB and XY, BC and YZ, CA and ZY), then the lines through their corresponding vertices (as AX, BY and CZ) are either parallel or concur

rent.

BOOK VII. POLYHEDRONS, CYLINDERS,

AND CONES

SECTION I. THE PRISM AND THE CYLINDER

97. Polyhedrons. A geometric solid bounded by planes is a polyhedron. The polygons that bound it are its faces, their intersections are its edges, and the intersections of its edges are its vertices; the faces taken together make up its surface. The area of its surface is the area of the polyhedron, and the amount of space that it occupies is the volume of the polyhedron. See also § 135, which explains how volume is measured.

A POLYHEDRON

A TETRAHEDRON

98. Number of Faces. It is evident that, in order to inclose space, a polyhedron must have at least three faces about each vertex, and at least four faces in all. A solid bounded by four faces is called a tetrahedron; by five faces, a pentahedron; by six faces, a hexahedron; by eight faces, an octahedron; by ten faces, a decahedron; by twelve faces, a dodecahedron; and by twenty faces, an icosahedron. The corresponding prefixes are used for the polyhedrons of any number of faces, but those here mentioned include the most common.