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. N 512 A plane is determined by certain conditions which fix its position. M -B 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 PROPOSITION I. THEOREM 514 A plane is determined by a straight line and a point without that line. M C A B N HYPOTHESIS. AB is a straight line and C is a point without AB. PROOF Let the plane MN pass through AB and turn on AB as an axis until it embraces C. Then the plane is determined; for if it now turns either way on AB, it will no longer contain the point C. § 512 Q. E. D. 515 COROLLARY 1. A plane is determined by three points. not in the same straight line. For let A, B, and C be the three points. Join AB. the line AB and the point C determine a plane (§ 514). Then 516 COROLLARY 2. A plane is determined by two intersecting lines. For let AB and AC be two intersecting lines. Then AB and any point in AC other than A determine a plane (§ 514). 517 COROLLARY 3. A plane is determined by two parallel lines. For let AB and CD be two parallel lines. M Then AB and CD are in the same plane (§ 114), and a plane containing AB and any point in CD is determined (§ 514). A B C D N PROPOSITION II. THEOREM 518 The intersection of two planes is a straight line. HYPOTHESIS. MN and PQ are two intersecting planes. PROOF § 511 § 514 Let the planes intersect in points A and B. Draw AB. Then AB lies in both planes. Any point without AB cannot be in both planes. Therefore the intersection of the two planes is the straight line AB. DEFINITIONS Q. E. D. 519 A straight line is perpendicular to a plane when it is perpendicular to every straight line in the plane drawn through its foot, that is, the point in which it meets the plane; and the plane is also perpendicular to the line. 520 A straight line and a plane are parallel if they cannot meet, however far they may be produced. 521 A straight line is oblique to a plane when it is neither perpendicular nor parallel to the plane; and the plane is also oblique to the line. 522 Two planes are parallel if they cannot meet, however far they may be produced. PROPOSITION III. THEOREM 523 If a straight line is perpendicular to each of two intersecting lines, it is perpendicular to their plane. M B N E HYPOTHESIS. AD is perpendicular to DB and DC. PROOF Let DF be any other line in the plane MN, and draw CB cutting DF in F. Produce AD to E, making DE = AD, and join A and E to C, F, and B. Since CD and BD are each L to AE at its middle point D, That is, AD is to DF, any line in the plane MN drawn through D. .. AD is to the plane MN. § 519 Q. E. D. |