NOTE ON THE TETRAHEDRON. 
[From the Oxford, Cambridge and Dublin Messenger of Mathematics, t. in. (1866), 
pp. 8—10.] 
The following simple properties of a tetrahedron seem worth noticing. 
In the tetrahedron ABCD if AC = BD and AD = BC, then the line joining the 
middle points of AB+CD, or say the points ^AB and | CD, cuts at right angles 
these lines AB and CD. 
If AB = CD, then the line joining the points \AC, \ BD, and the line joining 
the points \AD, \BC (lines which in any tetrahedron meet each other), cut each 
other at right angles. 
In fact if A, B, C, D have for their coordinates (a u ft, 7ft (a 2 , ft, y.i), (a 3 , ft, 73), 
(a 4j ft, 74): then the coordinates of the point ^ AB are £(ai + a 2 ), £(ft + ft 2 ), and so 
for the points CD, &c.: the equations of the line through the points \ AB, \ CD 
therefore are 
x-\ (a, + a 3 ) y - \ (ft + ft) _ z - (vi + 73) 
«1 + a, - a., - a 4 ft + ft - ft - ft 7i + 72 - 7s “ 74 ’ 
and I observe in passing that this line passes through the point whose coordinates are 
i ( a i + + a 3 + a i)> 4 (ft + ft + ft + ft), J (74 + 72 + 73 + 74); 
the other two similar lines pass through the same point, and the above-mentioned 
property of the general tetrahedron is thus proved. 
The condition that the foregoing line may cut at right angles the line AB, the 
equations whereof are 
a - «1 _ y~ ft _ z- 74 
«1 - «2 ft -ft 7i - 72 ’