99 
as. AT, BL,, Cl, concur at Q;. 
AY,, BY, CT, concur at Q,. 
AY,, BY,, CT concur at Q3. 
69. AB, DE, D,E, concur at x. 
BC, EF, E,F, concur at y. 
CA, FD, F,D, concur at z. 
x, y, 2 lie on a line n, say. 
70. AB, D,E,, D,#, concur at x3. 
BC, E,F;, E,F, concur at y,. 
CA, F,.D,, F,D, concur at z,. 
11, Yi, 2, lie on a line p. 
(ime as, NP ist; concur at a. 
BC ae, to J,;coneur.at 45. 
CA, QN, I,1, concur at z.. 
(N, P, Q are the feet of the interior angle bisectors. ) 
Loy Yo, 22 lie on a line q. 
72. The three lines n, p, g are concurrent. 
73. A’, B’, C’ are the midpoints of the sides of the triangle ABC. Lines 
drawn through A’, B’, C’, respectively, parallel to the triads of angular trans- 
yersals. which determine I, T,, T,, I, concur at I’, 1,%,,T,’ 3’. Then I’, 
r,T,’, TT’, 0;1,’ are concurrent at the centroid of the triangle ABC. 
74. IT’, 1,7,’, 1,0’, 131”, concur at the symmedian point of the triangle 
ABC. 
75. IQ, 1,@,, I,@2, I;Q, concur at the centroid of the triangle ABC. 
(The propositions 65 to 75 inclusive are taken from Mackay’s ‘‘ Euclid” and 
his ‘‘Symmedians and Concomitant Circles.’’) 
76. If DEF be the triangle formed by joining the inscribed points of contact 
of the triangle ABC; D,£, F, the triangle formed by joining the inscribed points 
of contact of the triangle DEF; D,E,F, the triangle formed by joining the 
inscribed points of contact of the triangle D, HZ, F,; J, I,, I,, I, are the inscribed 
and escribed centres. J, D, I, HE, I,F concur at the homothetic centres of the tri- 
angles DEF and I,1,1,. ID,, I,E,, I,F, concur at the homothetic centre of the 
triangles D, EH, F, and IJ,J,, and so on. (Dr. Mackay, Proceedings Edinburgh 
Math. Soc., Vol. I, pp. 51-2.) 
77. If three straight lines drawn from the vertices of a triangle are concur- 
rent, the three lines drawn parallel to them from the midpoints of the opposite 
sides are also concurrent; and the straight line joining the two points of concur- 
rency passes through the centroid of the triangle and is there trisected. (Frigier 
in Gergonne’s Annales, Vol. VII, 170.) 
