482 ON A THEOREM RELATING TO FIVE POINTS IN A PLANE. [364 
1. Considering the four summits 1, 2, 3, 4, which are such that no two of them 
belong to the same edge, then, if through any point 0 we draw 
the line OA' meeting the lines 41, 23, 
„ OB' „ „ 42, 31, 
* 00' „ „ 43, 12, 
and the lines Oa, 0/3, O7 parallel to the three edges of the cube respectively, the 
three planes (OA', Oa), (OB', 0/3), (OC', Oy) will meet in a line. 
2. For a properly selected position of the point 0, 
the lines OB', 00', Oa will lie in a plane, 
00', OA', 0/3 „ 
„ OA', OB', Oy „ „ 
In fact for such a position of 0, projecting the whole figure on any plane whatever, 
the lines 01, 02, 03, 04, OP, OQ, OR, OS, Oa, OS, O7, OA', OB', 00' meet the plane 
of projection in the points 1, 2, 3, 4, P, Q, R, S, a, /3, 7, A', B', C' related to each 
other as in the last-mentioned form of the plane theorem. To prove the two solid 
theorems, take 0 for the origin, Oa, OS, O7 for the axes, (a, S, 7) for the coordinates 
of the summit S, and 1 for the edge of the cube, 
the coordinates of 1 are a + 1, S, Y> 
,, 2 ,, a, /3 + 1, 7, 
„ 3 ,, a, S, 7 "i" 1, 
„ 4 „ a + 1, S + 1, 7 + 1. 
The equations of the line OA', or say of the line 0 (41, 23), are those of the planes 
0 41, 0 23, viz. these are 
CG 5 
y 
z 
= 0, 
oc, y 
z 
a + 1, 
s , 
1 
a, S+ 1, 
7 
a + 1, 
/3 + 1, 
7 + 1 
a, S 
7+1 
that is 
Writing for shortness 
these equations give 
x : y \ 
x(S-y) -(a-)rl)(y — z) = 0, 
x(S + 7 + 1) - a (y + z) = 0. 
M = OL + S + 7 + 1> 
2a (a +1) . 1 # 1 
(M + 27a) (M + 2a/3) ' M + 27a * M + 2 a/3 ’ 
or, completing the system, 
for line OA' we have 
2a (a+1) 
x ■ y : z -(M+2ya)(M+2aS) 
1 1 
M + 2 7 a : M + 2a/3 ’