89 
point of MK. The centre coincides with the centre of Brocard’s Circle. The 
radius is equal to 4 VR? + p2 where p is the radius of the Cosine Circle. The 
segments cut out of the sides of the triangle by the circle are proportional to the 
cubes of the sides of the triangle. For this reason the circle is sometimes called 
the Triplicate Ratio Circle. 
9. Taylor's Circle.—The circle which passes through the six projections of 
the vertices of the pedal triangle on the sides ot the fundamental triangle. 
10. Tucker’s Circles.—The circle that passes through six points determined 
as follows: If on the lines KA, KB, KC, points A’, B’, C’ are taken so that 
KA’: KA — KB’: KB= KC’: KC =a constant, then the six points above referred 
to are the intersections of B’C’, C’A’ and A’P’ with the sides of ABC. 
The centre is at the midpoint of the line joining M and the circumcentre of 
A’ BC’, 
The circum-, Lemoine, Cosine and Taylor Circles are particular cases of 
Tucker Circles. 
11.  Orthocentroidal Circle.—The circle of similitude of the circum and nine- 
points circle. Centre at the midpoint of HG. Radius is 3 HG. 
12. McCay’s Circles:—The three circles which circumscribe the triangles 
B,C,G, C,A,G, and A,B,G, respectively. (A,B,C, is Brocard’s second triangle 
and G is the centroid. 
13. Polar Circle.—This is the circle with respect to which the triangle is self- 
conjugate. Its centre is at H. It is real when H is outside the triangle, evanes- 
cent when H is at a vertex, and imaginary when H is within the triangle. 
14. 
B;, C;, which are the midpoints of AA’, BB’, CC’, respectively. (Proceedings 
Circle.—The circle on IM as diameter. It passes through A;, 
Indiana Academy of Sciences, 1898, page 89. ) 
15. Adam’s Circle.—The circle which passes through the six points determined 
by the intersection with the sides of the triangle of the lines through the Gergoume 
Point P, parallel to Ialb, Talc, Icla, respectively. The centre is at I. 
16. 
points), parallel to the sides of ThalipIc, IzalapI2c, and IsalspIsc, respectively, de- 
Circles.—Lines through P,, P,, P, (the associated Gergoume 
termine sets of six points on the sides which are concyclic. The centres of these 
three circles are at 1,, I,, and 1I,. These circles might be called the associated 
Adam’s circles. 
