94 
8. Lines drawn from the vertices to the points of contact of the escribed 
circles with the opposite sides concur at Q, Nagel’s Point. (For a number of 
interesting properties of this point, see Schwatt’s ‘‘Geometric Treatment of 
Curves.’’) 
9. Lines drawn from the vertices making equal angles with the sides AB, 
BC, CA, respectively, concur at 2 and 2’, the two Brocard points of the triangle. 
10. If A,, B,, C,, is Brocard’s first triangle, then AA,, BB,, CC), concur at 
D, the point isotomic conjugate to K. 
11. If L, M, N, be the midpoints of the sides of the triangle ABC, and L/, 
M’, N’ the midpoints of the sides of the triangle A, B,C,, then LL’, MM’ and 
NW’ concur at 8S. 
12. AI/, BM’, and CN’ concur at S’. S and S’ are isogonal conjugate 
points. (Schwatt’s ‘‘Geometric Treatment of Curves,” p. 5.) 
13. Perpendiculars from A, B, C upon B,C, C,A,, A, Bj, respectively, 
concur at N, a point on the circumcircle of the triangle ABC, known as Tarry’s 
Point. 
14. Lines through A, B, C, parallel to B,C,, C,A,, A, B,, respectively, con- 
cur at a point on the circumcircle of the triangle A BC known as Steiner’s Point. 
15. Parallels to AB and CA through Cand 8, respectively, concur with the 
median through A. There are evidently three such points of concurrency. These 
points are sometimes called the ezternal median points. 
16. If three lines through the vertices are concurrent, their isogonal conju- 
gates with respect to the angles of the triangle are also concurrent. (Steiner’s 
Gesammelte Werke I., 193, 1881.) If the ratios of the distances of the first point 
from the sides are / : m:n, those of the second point are 
oh ea 
Cn, th 
17. Perpendiculars to the sides of the triangle ABC from the midpoints of 
the sides of the orthic triangle of ABC are concurrent. (Edouard Lucas in Nou- 
velle Correspondance Mathématique II, 95, 218, 1876.) 
18. The ex-symmedians from any two vertices and the in-symmedian from 
the third vertex are concurrent. There are evidently three such points of 
concurrency. They are sometimes called the external symmedian points. 
19. If three lines drawn from the vertices of a triangle to intersect the oppo- 
site sides are concurrent, the lines isotomic conjugate to them are also concurrent. 
If the ratios of the distances of the first point of concurrency from the sides are 
1: m:n, the ratios of the second point are 
1 1 1 
a2l ° b2m~ c?n 
