98 
by Jacobi, but not explicitly stated by him. Lemoine stated it in 1873 to the 
Association Frangaise pour l’Avancement des Sciences. ) 
60. When three lines through the vertices of a triangle are concurrent, the 
six bisectors of the three angles they determine intersect with the corresponding 
sides of the triangle at six points, every three of which on different sides connect 
concurrently with the opposite vertices if an odd number of them is internal. 
61. When three points on the sides of a triangle are collinear, the six 
bisections of the three segments they determine connect with the corresponding 
vertices of the triangle by six lines, every three of which through different 
vertices are concurrent if an odd number of them is internal. 
62. When three points on the sides of a triangle are collinear, their three 
lines of connection with the opposite vertices determine an exscribed triangle 
whose vertices connect concurrently with those of the original to which they cor- 
respond. 
63. H,, H,, H, are points of intersection of AJ, BI, CI, respectively, with 
the inscribed circle. The perpendiculars from H,, H,, H, upon BC, CA, AB, 
respectively, are concurrent. 
64. The twelve radii from the incenter and the excenters of a triangle, per- 
pendicular to the sides of the triangle, meet by threes in four points which are the 
circumcenters of the triangles IZ, J,, IJ, J,,7,14,, 1,1, I. (J, 1,, I,, I, are 
the incenter and excenters. See note, p. 99, Vo]. I, Proceedings Edinburgh Math. 
Soe., Dr. Mackay). 
65. D, EB, Fare points of contact of J-circle with the sides of the triangle 
ABC, D,, E,, F, points of contact of J,-circle with sides, D,, E,, F,, of F,-circle, 
D,, E;, F, of I,-circle. 
AD,, BE,, CF, concur at T, 
AD,, BE,, CF, concur at T, 
AD,;, BE,, CF, concur at T, 
The points l',, l',, l', are called the associated Gergonne points. See No. 7. 
66. AD,, BE,, CF, concur at Q, 
AD,;, BE,, CF, concur at Q, 
AD,, BE,, CF concur at Q; 
Q,, Qo, Qs, together with Q given in No. 8, are called the Nagel points. 
67. AQ, BQ,, CQ, concur at T,. 
AQ;, BQ, CQ, concur at T,. 
AQ,, BQ,, CQ concur at T,. 
