107 
(26.) The intersections of the anti-parallel chords D! E, E! F, F! D with 
Lemoine’s parallels D E!, E F!, F D! respectively, are collinear. The 
D, E, F, D', E’, F', are the six points of intersection of Lemoine’s 
circle with the sides of the triangle. 
(Ibid, p. 182.) 
(27.) It the line joining two corresponding points of directly similar figures 
* F,, F,, F, described on the sides of the triangle ABC, pass through 
the centroid, the three corresponding points are collinear. 
(Ibid, p. 237.) 
(28.) If from Tarry’s point _|_’s be drawn to the sides B C, C A, A B of the tri- 
angle, meeting the sides in (a, a,, a,) (9, 3,, 8.) (Y, ¥1, Y2), the points 
a, B, y are collinear, so also (@,, 8,, y) and (a,, 8, y,). (Neuberg.) 
(Ibid, p. 241.) 
(29.) In any triangle AB C, O, O! are the centres of the inscribed circle and of 
the escribed circle opposite A; OO! meets BCin D. Any straight line 
through D meets AB, AC respectively in b, ec. Ob, O' e intersect in 
P, O1b, OcinQ. PA Qis a straight line perpendicular to O O!. 
(Wolstenholme—Math. Problems, p. 8, No. 79.) 
(30.) A triangle P QR circumscribes a circle. A second triangle A BC is 
formed by taking points on the sides of this triangle such that A P, 
BQ, C Rare concurrent. From the points A, B, C tangents A a, B b, 
C c aredrawn tothe circle. These tangents produced intersect the sides 
BC, C A, AB, in the three points a b c, which are collinear. 
(Catalan Geométrie Elementiaire, p. 250.) 
(31.) The three internal and three external bisectors of the angles of a triangle 
meet the opposite sides in six points which lie three by three in four 
straight lines. 
(Richardson and Ramsey—-Modern Plane Geometry, p. 19.) 
(32.) If O be any point, then the external bisectors of the angles BOC, CO A, 
A OB meet the sides BC, C A, AB respectively in three collinear 
points. 
(Ibid, p. 52.) 
(83.) The external bisectors of the angles of a triangle meet the opposite sides 
in collinear points. (A special case of 31.) 
(Lachlan—Modern Pure Geometry, p. 57.) 
(34.) Lines drawn through any point O perpendicular to the lines O A, O B, OC 
meet the sides of the triangle AB C in three collinear points. 
(Ibid, p. 59.) 
