111 
(63.) The feet of the perpendiculars on the sides of a triangle from any point in 
the circumference of the cireumcircle are collinear. (Simson’s line.) 
(64.) If two triangles are in perspective the intersections of the corresponding 
sides are collinear. A different statement of 49. 
(Mulcahy—Modern Geometry, p. 23.) 
(65.) The perpendiculars to the bisectors of the angles of a triangle at their 
middle points meet the sides opposite those angles in three points which 
are collinear. 
(G. DeLong Champs.) (Mackay, Euclid, p. 356.) 
I, I,, I., I, are the centres of the inscribed and three escribed circles of the tri- 
angle A BC. D, E, F; D;, E,, F,; D,, E,, F,; Dz; E,; F,: are the 
feet of the perpendiculars from these centres upon the respective sides. 
N, P, Q are the feet of the bisectors of the angles A, B, C. 
(66.) AB, DE, D, E, concur at Q). 
BC, EF, E; F, concur at Nj. 
CAC FH DLE D, coneur at P;. 
Q,, N,, and P, are collinear. 
(67-)) A.B, D, E,, D,'E, concur at Q,. 
BC) BE, EF.) 8; BF; concur at No. 
CRAG aD Lae sconcur atk. 
Q., N,, and P, are collinear. 
(68.) A B, N P, I, I, concur at Q,. 
BC PO =) rconcurat N5. 
CrAQ IN. to i concuriat 2, : 
Q, N, and P, are collinear. 
(66, 67, 68—Stephen Watson in Lady’s and Gentleman’s Diary for 1867, p, 72. 
Mackay, Euclid, p. 357.) 
(69.) Ma, the middle point of Q O and the middle point of Q A are collinear. 
(Mackay, Euclid, p. 363.) 
(70.) The six lines joining two and two the centres of the four circles touching 
the sides of the triangle A BC, pass each through a vertex of the tri- 
angle. 
(Mackay, Euclid, p. 252.) 
(71.) Ma, O, and the middle of the line drawn from the vertex to the point of 
inscribed contact on the base are collinear. A similar property holds 
for the escribed centres. 
(Mackay, Euclid, p. 360.) | 
InpranaA UNIVERSITY, 
December 18, 1898. 
