(45.) 
(46. ) 
(47.) 
(48.) 
(51.) 
(52.) 
(53.) 
109 
AB C is a triangle inscribed in and in perspective with A’ B' C?; the tan- 
gents from A BC to the incircle of A! B! C! meet the opposite sides 
in three collinear points, X, Y, Z (BC in X, etc.). 
(Ibid, p. 128.) 
If three pairs of tangents drawn from the vertices of a triangle to any 
circle, meet the opposite sides X, X!, Y, Y!, Z, Z', and if X, Y, Z are 
collinear, so also are X!, Y?, Z?. 
(Ibid, p. 128.) 
Ii X Y Z is a transversal and if X!, Y! Z! are the harmonic conjugates of 
X,Y, Z, then 
ts Zhe BOR xk are collinear. 
Also the middle points of X X!, Y Y', Z Z! are collinear. 
(Ibid, p. 131.) 
Tf L is an axis of symmetry to the congruent triangles A B C and A! B! CG! 
and O is any point on L, A? O, B! O, C1 O cut the sides BC, C A, AB 
in three collinear points. 
(Depuis—Modern Synthetic Geometry, p. 204.) 
Two triangles which have their vertices connecting concurrently, have 
their corresponding sides intersecting collinearly. 
(Desargue’s Theorem.) (Ibid, p. 204.) 
A!, B,? C! are points on sides of AB C such that AA!, BB!, C C! are con- 
current, then AB, A’ B'; BC, B! C!, C A, C! A? meet in three points 
Z, X, Y which are collinear. 
(Ibid, p. 205.) 
If P be any point, AB C a triangle and A’ B! C? its polar reciprocal with 
respect to a polar centre O, the perpendiculars from O on the joins P A, 
P B, and P C intersect the sides of A! B! C? collinearly. 
(Ibid, p. 223.) 
If the three vertices of a triangle be reflected with respect to any line, the 
three lines connecting the reflexions with any point on the line intersect 
collinearly with the opposite sides. 
(Townsend—Modern Geometry, p. 180.) 
When three of the six tangents to a circle from three vertices of a triangle 
intersect collinearly with the opposite sides, the remaining three also 
intersect collinearly with the opposite sides. 
(Ibid, p. 180.) 
