A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

216 A SEQUEL TO EUCLID.
cos B, cos, C, &c. The circle in this ease is that of
Prop. 9, and is called the COSINE, CIRCLE Of the _polygon.
20. If I = m, the line OK will be bisected in 0', and
the circle will be concentric with the Brocard circle.
This is, by a~nalogy,,called the LEmoiNE CirCLE, of t h e
polygon: its Fuis i'> Ia to li sec (o, and the intercept
which it makes on the sides are proportional to sin (A - o),
sin (B - 0)), ~-c.
30* If I = m tan e), the intercepts are proportional to
sin (A -7.n/4), sin (B - 7-/4), sc., and the radius is equal
RI sin w cosec (0) ~ r/
40. If I = m tan2 co. The centre of the circle is the
middle point of the line O2&Y, its radius is equal to iRsin o,
and the intercepts are proportional to cos (A~ 0), cos (B + o),!~c. These will be the _projections of MY7~ on the sides of the
_polygon.
5'* If the polygon reduce to a triangle, and the ratio
of I1: m be
- cos AcoslB cos C: Il+cos AcosiB cos C,
the intercepts are, respectively, equal to
~R sin 2A cos (B - C),
~3 sin 2B cos (C - A),
II sin 2C cos (A -B).
The perpendiculars from the centre on the sides will be
proportional to cos' A, cos' B, cos2 C. This is the case of
Taylor's Circle. The ratio I1: m expressed in terms of
0) is sin (A - co) - sin'A: -sin' A.
6'. Any Tucker's circle of the triangle ABC is a Taylor's
circle of some other triangle having the same circumcircle
and symmedian point.
For the Tucker's circle of the triangle ABC being
given, the ratio I1: m is given, and from the proportion
sin (A - w) - sin3A: sin3A I1: m, we get, putting cot
A = x, the equation
XI - cot W. x' + X + (1 + 1/rn) cosec ce - cot (O = 0;
the three roots of which are the cotangents of the three
angles of the required triangle.