Elements of descriptive geometry, with applications to isometric projection and othe forms of one-plane projection; a text-book for colleges and ingineering schools by O. E. Randall.

RELATING TO POINT, LINE, AND PLANE

53

taking the position n"1 and its horizontal projection taking the
position n,. The distance m,-n,, is the distance sought.
Check. The result obtained by use of any one of the methods
here suggested may be checked by using one of the other methods.
135. Problem 73. Find the distance
nl, — Kn,
between the two points l= - 6, - 6, 2 I
and N= 6, 2, - 4 by Analysis 2.
136. Problem 74. Find the distance G '
between the two points M =- 6, 6, 2 and
N= 4, - 4, - 4 by Analysis 3. / \
137. Problem 75. Find the distance \
between the two points M= O, - 6, 1 and n
N= 0, 2, 8 by any method, and check
it by another.
138. Problem 76. To pass a plane through three given points.
Analysis. If we connect any two of the given points by a straight
line, such a line must lie in the required plane and must pierce H
and V in the corresponding traces of the plane (see Section 42).
A straight line connecting either one of the points just used with
the remaining point must
also lie in the required plane
and therefore must pierce H
and V in the corresponding
~/ i ~\ '. traces of the required plane.
/^/,' \ A straight line obtained
/i /o:', \o/ by joining any one of the
G-ut_-ii — -L three given points with any
point in a straight line con\^^ \, j necting the other two, or
\-~~ \,! by drawing through any
one of the three points a
F. 63 straight line parallel to the
rIo. 63
straight line connecting the
other two, is a line of the required plane.
Construction. See Fig. 63. Let M, N, and 0 represent the three
given points. Connect M and N by a straight line and produce it
to intersect H in a, and V in b. The point a, is a point in the