= 
105 
Q is Nagel’s point. (It is the point of concurrency of the three lines joining 
the vertices to the points of tangency of the three escribed circles.) 
Q! is the isotomic conjugate of Q. 
O is the centre of the inscribed circle. 
S is the point of perspective of Ma Mp Me and Mia Mwy Mic. > 
S' is the point isogonal conjugate to S. 
R is the point of concurrence of perpendiculars from A, B, C on the sides of 
Nagel’s triangle. 
M, is the centre of Nagel’s circle. 
T is the point of concurrence of perpendiculars from A’ B’ (” upon the re- 
spective sides of A’” B” C0”. 
Z is Brocard’s centre. 
Z' is the point isogonal conjugate to Z. 
P is the point isotomic conjugate to O. 
P' is the point isogonal conjugate to P. 
Q, is the point isogonal conjugate to Q'. 
F is the centre of Nine points circle. 
THEOREMS. 
The original sources of the theorems are known in only a very few cases. The 
references simply indicate where the theorems may be found. 
(1.) M, H and G are collinear. 
(Lachlan—Modern Pure Geometry, p. 67.) 
(2:) K, G and the Symmedian point of Ma My Me are collinear. 
(Ibid, p. 138.) 
(3.) Tangents to the circumcircle at the vertices of AB C form the triange P Q 
R; Ha, Hv, He are the feet of the altitudes of AB C; P Ha, QHb, 
RH¢ are concurrent in a point which is collinear with M and H. 
(Ibid, p. 138.) 
(4.) M, K and the orthocentre of its pedal triangle are collinear. 
(McClellan—The Geometry of the Circle, p. 83.) 
(5) M and the orthocentre of its pedal triangle are equidistant from and col- 
linear, with the centre of Taylor’s Circle. 
(Ibid, p. 83.) 
(6.) Q, Q! and P are collinear. 
(Aley—Contributions to the Geometry of the Triangle, p. 8.) 
(7.) K, P! and Q, are collinear. 
(Ibid, p. 13.) 
