97 
47. lf X, Y, Zbe three points on the sides of the triangle ABC such that 
the pencil D(AC, EF) is harmonic, then AD, BE, CF are concurrent. 
48. lf tangents to the cireumcircle at the vertices of the triangle ABC, meet 
in L, M, N, then AL, BM and CN are concurrent. 
49. If on the sides of the triangle ABC, similar isosceles triangles LBC, 
MCA, NAB be described, AL, BM, CN are concurrent. 
50. If the ex-citeles touch the sides to which they correspond in D,, E,, F3, 
the perpendiculars to the sides through these points are concurrent. 
51. If D, EH, Fare the points of contact of the incircle with the sides of the 
triangle ABC and if DJ, EI, FI meet EF, FD, DE in L, M, N, respectively, then 
AL, BM, CN concur. 
52. If DD’, EE’, FF’ are diameters of the incircle through D, E, F, the 
points of contact with the sides of the triangle ABC, then AD’, BE’, CF’ concur. 
53. If P, Q, R be collinear points in the sides BC, CA, AB of the triangle 
ABC, and if P’, Q FR’ be their harmonic conjugates with respect to those sides 
then AP’, BQ’, CR’ are concurrent. 
54. If squares APQB, BUVC, CXYA be described upon the sides of the 
triangle ABC (all externally or all internally) and if QP meet XY in a, PQ 
meet VU in 8, UV meet YX in y, then aA, 8B, yC concur in K the symmedian 
point. (Halsted, “El. Synthetic Geometry,” p. 150.) 
55. A’B’C” is the pedal triangle of 2, and A’ B’C” is the pedal triangle 
of 2%, B’ CU’, C” A’, A” B’ form the triangle X YZ, whose sides are parallel 
to the sides of ABC. PQR is the pedal triangle of ABC. PX, QY, RZ concur 
at the cireumcenter of X YZ. 
56. The Simson lines of the median triangle LMN of the triangle ABC, 
with respect to the vertices P, Q, R of the pedal triangle, concur at the center of 
Taylor’s circle. 
57. The Simson lines of the pedal triangle PQR of the triangle ABC, with 
respect to the vertices L, WV, N of the median triangle concur at the center of 
Taylor’s circle. 
58. If BW, CV be perpendicular to BC; CU, AW perpendicular to CA; 
AV, BU perpendicular to AB; then AU, BV, CW concur at the circumcenter 
of ABC. (C.F. A. Jacobi, ‘De Triangulorum Rectilineorum Proprietatibus,” 
p. 56.) 
59. If triangles A, B,C, and A,B,C, are circumscribed about the triangle 
ABC in such a manner that their sides are perpendicular to those of ABC, 
then A,A,, B, B,, C,C, concur at the circumcenter of ABC. (Probably known 
7—ScIENCR. 
