Descriptive geometry . to the ground line (§ 102), onepoint on each trace being located by the line B. Special Case III. Suppose both given lines parallel to theground line. Let A and B, Fig. 139, be the given lines; thenHQ and VQ are both parallel to GL. To find a poinl on each,we may find first the profile trace of the plane Q on anyassumed profile plane; this trace, PQ, will pass through theprofile traces up and , of A and B respectively. 82 DESCRIPTIVE GEOMETRY [X, § 106 Corollary I. To find the plane which contains a given lineand a given point. Analysis. Through the given point, draw
Descriptive geometry . to the ground line (§ 102), onepoint on each trace being located by the line B. Special Case III. Suppose both given lines parallel to theground line. Let A and B, Fig. 139, be the given lines; thenHQ and VQ are both parallel to GL. To find a poinl on each,we may find first the profile trace of the plane Q on anyassumed profile plane; this trace, PQ, will pass through theprofile traces up and , of A and B respectively. 82 DESCRIPTIVE GEOMETRY [X, § 106 Corollary I. To find the plane which contains a given lineand a given point. Analysis. Through the given point, draw an auxiliary lineparallel to the given line, thus reducing the problem to one oftwo parallel lines. Or, assume any point on the given line,and draw an auxiliary line through this point and the givenpoint, thus reducing the problem to two intersecting lines. Thefirst method is the one usually adopted. Construction (Fig. 140). A is the given line, c the givenpoint. The auxiliary line B is drawn through c parallel to A. Fig. 140. (§ 93), and the required plane Q is passed through the linesA and B. In Fig. 141 the line A is parallel to V\ hence the directionof VQ is known to be parallel to Av (§ 102). But as neitherA nor B has a F-trace, a point on VQ is determined by notingwhere HQ intersects GL. Corollary II. To find the plane which contains three givenpoints not in the same straight line. Analysis. Connect any two points by a straight line, andthrough the third point draw a second line parallel to the first, X, § 107] PARALLEL LINES AND PLANES 83 thus reducing the problem to two parallel lines. Or, drawlines connecting any point with each of the other two, thusreducing the problem to one of two intersecting lines.
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