(56.) 
(60.) 
(61.) 
(62. ) 
If from the middle points of the sides of the triangle A B C, tangents be 
drawn to the corresponding Neuberg circles, the points of contact lie on 
two right lines through the centroid of A BC. 
(Casey—Sequel, p. 241.) : 
If P is a Simson’s point for A BC, and O any other point on the circum- 
circle of A BC, then the projections of O upon the Simson’s lines of O 
with respect to the triangles P A C, P BC, PC A, A BC are collinear. 
(Lachlan—Modern Pure Geometry, p. 69.) 
When three lines through the vertices of a triangle are concurrent, the six 
bisectors of the three angles they determine intersect the corresponding 
sides of the triangle at six points, every three of which on different 
sides are collinear if an odd number is external. 
(Ibid, p. 181.) 
When three points on the sides of a triangle are collinear, the six bisec- 
tions of the three segments they determine connect with the correspond- 
ing vertices of the triangle by six lines, every three of which through 
different vertices are collinearly intersectant with the opposite sides if 
an odd number is external. 
(Ibid, p. 182.) 
A,, Mia and K?, the intersection of the Symmedian through A and the 
tangent to circumcircle at C, are collinear. 
(Schwatt—Geometric Treatment of Curves, p. 4.) 
If M X and F Y are parallel radii, in the same direction, in circumeircle 
and Feuerbach circle, then X, Y, and H are collinear. 
(Ibid, p. 21.) 
If Y, is the other extremity of the diameter F Y, then Y,, G, and X are 
collinear. 
(Ibid, p. 21.) 
If P, a point on the circumcircle of ABC be joined with H', H'', H'?, 
the respective intersections of the produced altitudes with circumcircle, 
and if the points of intersection of P H', P H'?, P H'!? with BC, CA, 
AB be U, V, W respectively, then U, V, W are collinear. 
(Ibid, p. 23.) 
A O, BO, CO meet the circumcircle in A', B', C'; perpendiculars from 
M upon the sides BC, C A, AB meet Nagel’s circle in A'?, B'?, C1?; 
the corresponding sides of A! B'C! and A‘! B'! C!! meet in three 
collinear points. . 
(Ibid, p. 40.) 
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