A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

104 Small Circles on the Sphere.
97. If the secant through a centre of similitude S meets the
circles in the homothetic points XM, M; then tan ~ SM: tan SNM'
in a given ratio.
DEM.-From the definition
sin SP: sin PM:: sin SP': sin P'f.
Hence it follows that the angle SMIP = SM'P'.
Now since the triangles SMP, 3SM'P' have two angles in one
respectively equal to two angles in the other, it follows from
the third of Napier's Analogies that
tan: SM: tan 1 SM:' tan 1 (SP +P P) tan 1 (SP '+P'1T);
that is, in a given ratio. Similarly,
tan L S N: tan ' SN' in a given ratio. (398)
Cor.tan 1 SM f. tan 1 SN' is constant, as also tan i SN. tan 1 SM'.
This follows from ~~ 92, 97. (Compare Sequel, Prop. II.,
page 83.
Cor. —If there be given a point S and a circle Y, and on
the arc Sf joining S to any point MJon Y a point N' be taken,
such that tan 2 S1. tan 1 S7V' is constant, the locus of N' is a
circle.
98. The six centres of similitude of three small circles taken in
pairs lie three by three on four great circles, called axes of similitude of the small circles.
DEM.-If a, b, c be the radii of the circles; A, B, C their
spherical centres, A'B'C' the internal centres of similitude,
and A"B"C" the externals; then we have by definitions
(AB, C") = a - b, (BC, A")= b — c, (CA, B") = c a.
Hence (AB, C"). (B C, A"). ( CA, B") = 1.
Hence (~ 70) the points A", B", C" lie on a great circle.
Similarly, it may be shown that any two internal centres and
an external centre lie on a great circle.