507 A figure is symmetrical with respect to a center, when every line drawn through the center cuts the perimeter in two points symmetri cal with respect to the center. 508 The straight line drawn through the center of a symmetrical figure, and terminated by the perimeter, is called a diameter. PROPOSITION VIII. THEOREM 509 If a figure is symmetrical with respect to two axes perpendicular to each other, it is also symmetrical with respect to their intersection as a center. HYPOTHESIS. The figure ABCDEFGH is symmetrical with respect to the perpendicular axes XX' and YY' intersecting at O. CONCLUSION. ABCDEFGH is symmetrical with respect to 0. PROOF Let P be any point in the perimeter. Draw PLQ parallel to XX', and QKP' parallel to YY'. Join LK, OP, and OP'. That is, the figure is symmetrical with respect to O as a center. EXERCISES § 507 Q. E. D. 1116 A circle is symmetrical with respect to its center, or with respect to any diameter as an axis. 1117 A parallelogram is symmetrical with respect to the intersection of its diagonals as a center. 1118 Every regular polygon of an even number of sides has a center of symmetry. 1119 An isosceles triangle is symmetrical with respect to the altitude upon the base. 1120 An equilateral triangle is symmetrical with respect to all of its altitudes. 1121 The symmetrical of a straight line, with respect to an axis, or with respect to a center, is an equal straight line. 1122 The symmetrical of an angle with respect to a center is an equal angle. 1123 The symmetrical of an angle with respect to an axis is an equal angle. 1124 If two straight lines are symmetrical with respect to a center, they are equal, parallel, and extend in opposite directions. 1125 If two polygons are symmetrical with respect to a center, or with respect to an axis, they are equal. 1126 A trapezium has no center of symmetry. 1127 An isosceles trapezoid has an axis of symmetry. 1128 Each diagonal of a square is an axis of symmetry. 1129 How many axes of symmetry has each of the regular polygons considered in this book? 1130 How many axes of symmetry has a regular polygon of 2 n sides? of 2 n + 1 sides? SOLID GEOMETRY BOOK VI STRAIGHT LINES AND PLANES DEFINITIONS 510 Solid Geometry treats of figures whose parts are not all in the same plane. M 511 A plane is a surface such that the straight line which joins any two of its points lies wholly in the surface (§ 54). A plane is understood to be indefinite in extent. We may represent a plane by a parallelogram drawn in perspective and lying in the plane; as, the plane MN. 512 A plane is determined by certain conditions which fix its position. M 513 Postulate. A plane may be revolved about any line. lying in it. Thus, the plane MN may be revolved about the straight line AB as an axis until it passes through any fixed point in space. Hence any number of planes may pass through, or embrace, a straight line; that is, one straight line does not determine a plane. A N -B |