[964 
964] ON THE NINE-POINTS CIRCLE OF A SPHERICAL TRIANGLE. 
551 
the half-sides 
the triangle 
diculars from 
ntre of the 
Lave 
B'G', C'H' 
Secondly, for the points L', M', N', we have 
sin B'L' = 
rpi 
V(? 2 + r 2 + 2pqr) ’ 
sin C'L' = 
qpi 
V(? 2 + r 2 -f 2pqr) ’ 
sin B'L' : 
: sin C'L' = r 
: q, 
= cos B'A' : 
: cos C'A' 
sin C'M' : 
sin A'M'= p 
: r, 
= cos C'B' 
: cos ri'#, 
sin ri W' : 
sin B'N' = q : 
: p, 
= cos A'C' 
: cos B'C', 
that is, 
and similarly 
viz. the sides B'C', C'A', A'B' are by the points L', M', N' divided each into two 
parts such that for any side the sines of the two parts are proportional to the cosines 
of the other two sides. We have 
sin B'L'. sin C'M'. sin A'N' — sin C'L'. sin A'M'. sin B'N', 
viz. the arcs A'L', B'M', C'N' meet in a point K' which may be called the cos-centre 
of the triangle A'B'G' (where observe that, for a, b, c indefinitely small, i.e. for a plane 
triangle, the points L', M', N' are the mid-points of the sides, and the centre K' is 
the c. G. or median point of the triangle). 
But further, we have 
and thence 
sin 2B'L' (= sin BL) = 2rpl (q , 
(p + r 2 + 2pqr 
sin 2C'L' (= sin CL) = 2qPl (r +Pq) , 
7 (f + r- + 2pqr 
sin BL . sin C'ilf. sin AN = sin CL . sin AM. sin BN, 
viz. the arcs AL, BM, CN meet in a point, which is obviously not the cos-centre of 
the triangle ABC. 
We have thus the construction of the nine-points circle as a six-points circle, by 
means of the points F, G, H, L, M, N; and by way of recapitulation we may say that 
the nine-points circle meets the sides BC, CA, AB in the points F, L; G, M ; H, N 
respectively, where the points F, G, H depend on the ortho-centre of the semi-triangle, 
and the points L, M, N depend on the cos-centre of the semi-triangle. 
The triangle ABC has an inscribed circle and three escribed circles, and we have 
(as is known) the theorem that the nine-points circle touches each of these four circles. 
The circles BC, CA, AB and the nine-points circle form a tetrad of circles, and the 
inscribed circle and the three escribed circles a tetrad of circles, or say the eight 
circles form a bitetrad, such that each circle of the one tetrad touches each circle of 
the other tetrad. 
orthocentre