BOOK VII. ON PLANES, AND LINES IN SPACE. SECTION I-MISCELLANEOUS PROPOSITIONS. PROPOSITION 1. It is always possible to find one plane, and only one plane, containing a given straight line and a given point not in that straight line. A Fig. 1. Let BC be the given straight line and A the given point not situated in the straight line BC, then it shall be always possible to find one plane, and only one plane, containing the straight line BC and the point A. B H F E D C Suppose a plane of indefinite extent to revolve round the straight line BC, then since in the course of its revolution such a plane must pass through every point in space, therefore there must be some one position of this plane in which it passes through the given point A. That is, one plane may always be found containing the given straight line BC and the given point A. Also, no more than one such plane can be found. If possible, let ABCD and ABCE be two non-coincident planes, each of which contains the straight line BC and the point A. Take any point F in BC. Because each of the points A and F is situated in the plane ABCD, therefore a straight line AGF may be drawn from A to F, and situated in the plane ABCD (Def. 9). Similarly, a straight line AHF may be drawn from A to F, and situated in the plane ABCE. Therefore the two non-coincident straight lines AGF and AHF intersect in two points A and F, which is impossible (Ax. 1); therefore the planes ABCD and ABCE must coincide in every point. Corollary 1.-One plane, and only one plane, may always be found containing three given points not situated in the same straight line. Corollary 2.-One plane, and only one plane, may always be found containing two given intersecting straight lines. PROPOSITION 2. It is always possible to find one plane, and only one plane, containing two given parallel straight lines. Let AB and CD be the two given parallel straight lines, then it shall be always possible to find one plane, and only one plane, containing both of the straight lines AB and CD. Because AB and CD are parallel straight lines, therefore they are situated in one plane (Def. 27); A C Fig. 2. B E that is, one plane may always be found containing both AB and CD. Also no more than one such plane can be found. For let E be any point in CD. D Because E is a point not situated in the straight line AB, therefore only one plane can be found containing the straight line AB and the point E (Bk. VII. Prop. 1); R 2 therefore only one plane can be found containing the two given parallel straight lines AB and CD. Note. It follows from the two preceding propositions that a plane is completely determined in each of the three following cases: 1. When it contains a given straight line and a given point not in that line. 2. When it contains three given points not situated in the same straight line. 3. When it contains each of two given intersecting or parallel straight lines. PROPOSITION 3. Through any given point it is always possible to draw one straight line, and one straight line only, parallel to any given straight line not containing the given point. Let A be the given point, and BC the given straight line not containing the point A, then it shall be always possible to draw one straight line, and one straight line only, through the point A parallel to the straight line BC. Fig. 3. B A C D Because the point A is not situated upon the straight line BC, therefore one plane, and one plane only, may always be found containing the straight line BC and the point A. Because (Ax. 4 and Bk. III. Prob. 6) it is always possible to draw one straight line, and one straight line only, in the plane containing the straight line BC and the point A, and parallel to BC, let this straight line be AD; therefore AD will be drawn through the given point A, parallel to the given straight line BC, and will be the only straight line that can be so drawn. PROPOSITION 4. If two planes cut one another their line of intersection shall be a straight line. Fig. 4. Let ABC and ABD be two planes which cut one another, and let A and B be two points in their line of intersection, then if AB be joined, every point common to both of the planes shall be situated in the straight line AB or in AB produced. If possible let P be a P A B D point in the line of intersection of these planes and not situated in the straight line AB or that line produced. Because A, P, and B are three points not situated in the same straight line, therefore no more than one plane can be found which contains the points A, P, and B. Because A, P, and B are three points situated in the line of intersection of the two planes ABC and ABD, therefore there are two planes ABC and ABD, each of which contains the three points A, P, and B. But we have before proved that only one plane can be found containing these three points, which is impossible; therefore every point in the line of intersection of the planes ABC and ABD must be in the straight line AB; that is, the line of intersection of these planes is a straight line. DEFINITIONS. 80. A straight line and a plane are said to be parallel to one another when they cannot meet, however far they may be produced in any direction whatever. 81.-Two planes are said to be parallel to one another when they cannot intersect, however far they may be produced in any direction whatever. PROPOSITION 5. If a given straight line be parallel to a given plane, every plane which contains the given straight line shall intersect the given plane in a straight line parallel to the given straight line. Let AB be the given plane and CD the given straight ever far they may be produced in any direction whatever, therefore CD and EF cannot meet, however far they may be produced in either direction. But CD and EF are by hypothesis in the same plane, therefore EF is parallel to CD (Def. 27). PROPOSITION 6. If a given straight line be situated in a given plane, every line parallel to the given straight line shall either be situated in the given plane or be parallel to that plane. Take the same figure as in the last Proposition. Let EF be the given straight line and AB the given plane in which EF is situated, and let CD be a straight line |