SECTION I. INTERSECTIONS AND PARALLELS. The following postulate is usually tacitly assumed in elementary geometry, but may with advantage be explicitly stated. POST. 1. Any assemblage of points, lines, surfaces, or solids may have its position in space changed without any change in the relative positions of its parts. Or, (using the term figure for such assemblage) Any figure may be transferred from one position in space to another without change in shape or size. Def. 1. A plane is a surface such that, if any two points are taken in it, the straight line passing through these points lies wholly in the surface. A plane must be understood as indefinitely extended in all directions, as are the straight lines which lie in it. A limited portion of a plane forms a plane figure. Def. 2. Straight lines which are in the same plane and do not meet (however far extended) are said to be parallel. Def. 3. A straight line, which does not meet a plane (however far extended), is said to be parallel to the plane. Def. 4. Planes, which do not meet (however far extended), are said to be parallel. THEOREM I. (including Euc. XI. 2). A plane is determined by either (1) three points, not in the same straight line; or (3) two straight lines which either meet or are parallel. Let A, B, C be three points not in the same straight line. Then any plane may be so changed in position as to pass through A, next moved about A fixed until it passes through B, and finally rotated about A and B fixed until it passes through C; it is then fixed in position, and, being completely determined by the points A, B, C, may be denoted as the plane ABC. Also, since the straight line through A and B lies in the plane, (Def. 1.) the plane is completely determined by the line through AB and the point C, and may be described as the plane through the line AB and the point C. B Further, since the straight line through A and C lies in the plane, the plane is completely determined by the two straight lines AB, AC intersecting at A, and may be described as the plane through the lines AB, AC. Lastly, if two lines are parallel, they are (by Def.) in the same plane, and that plane is determined when the lines are given, so that it may be described as the plane through the two parallel lines. Q. E. D. COR. I. A triangle is the only rectilineal figure that is necessarily plane. COR. 2. A plane may be generated by a straight line, which always passes through a fixed point and intersects a fixed straight line which does not pass through that point. Also by a straight line which always passes through two fixed straight lines, which intersect or which are parallel. THEOREM II. (including Euc. XI. 3). If a plane and a straight line meet, they intersect in one point, and if two planes meet, they intersect in one straight line. For, if a straight line through a point A without a plane meet that plane in the point B, it cannot meet it again, since, if it did, it would lie wholly in the plane. (Def. 1.) Also, if A, B are two points in the line of intersection of two planes, the straight line through A and B lies in both planes, (Def. 1.) and any other point C in their line of intersection must be in this line, for otherwise the two planes would coincide: (Th. I.) hence the planes intersect in one straight line. Q. E. D. THEOREM III. If three planes intersect each with the other two, the three lines of intersection either meet in one point or are parallel. Let A, B, C be three planes; then the lines of intersection of B and C, of C and A, and of A and B either meet in a point or are parallel. If the intersection of B and C meet the plane A in 0: then, since O is in both the planes A and B, the intersection of A and B passes through 0; and since is in both the planes A and C, the intersection of A and C passes through 0: therefore the three lines of intersection meet in O. If the intersection of B and C is parallel to the plane A: then, supposing that the intersections of A and B and of A and C met in the point 0, O would be a point in the intersection of B and C, so that the intersection of B and C would meet the plane A: hence the intersections of A and B and of A and C do not meet, and therefore, being in the plane A, they are parallel : (Def. 2.) also, being in the plane A, they cannot meet the intersection of B and C, which is parallel to that plane : therefore the three intersections in this case are parallel. Q. E. D. COR. If two intersecting planes pass through two parallel lines, their intersection is parallel to these lines. THEOREM IV. (Euc. XI. 16). The intersections of a plane with two parallel planes are parallel. For these intersections do not meet, since, if they did, the two planes would meet; whereas, being parallel, they do not meet. Further, they are in the same plane, therefore they are parallel. (Def. 2.) Q. E. D. |