100 
78. If ABC be any triangle and O any point whatever, and A,, B,, C, be 
points symmetrical to O with respect to the midpoints of BC, CA, AB, then 
AA,, BB,, CC, concur ata point P. The centroid G lies on the line OP and 
divides it in a constant ratio. (M.d’Ocagne in Nouvelles Annales, Third Series 
I, 239.) 
79. It through K (Grebe’s Point) parallels to the sides BC, CA, AB of the 
triangle ABC are drawn, meeting these sides in D, D’; E, E’; F, F’, respectively, 
and if EF and E’F’ intersect inp; FD and F’D’ ing; DE and D’E’ int, then 
Ap, Bq, Cr are concurrent. (Dr. Mackay, ‘‘Symmedians of the Triangle,” etc., 
p- 39.) 
80. A’, B’, C’ are the midpoints of the sides of the triangle ABC, and 
I, I,, I,, I;, are the in and ex centers. 
I, A’, I, B’, I,,C’ concur at the symmedian point of the triangle J, I, J;. 
IA’, I, B’, I,C’ concur at the symmedian point of the triangle J J, /,. 
I, A’, IB’, I,C’ concur at the symmedian point of the triangle J, I/,. 
I,,A, I, B’, IC’ concur at the symmedian point of the triangle I, 7, I. 
81. If AK, BK, CK cut the sides of the triangle ABC at the points R, S, T 
and the circumcircle of the triangle ABC at the points D, E, F, then 
AK, BF, CE are concurrent. 
BK, CD, AT are concurrent. 
CK, AE, BD are concurrent. 
82. X, Y, Z are the feet of the perpendiculars in the triangle ABC. If 
H,, H,, H, be the ortho-centers of the triangles AYZ, ZBX, XYC, then the 
lines H, X, H, Y, H,Z are concurrent. 
83. If H,’, H,’, H,’ be the ortho-centers of the triangles HYZ, XCZ, XYB. 
H,”, H,’’, H,/ be the ortho-centers of the triangles CYZ, X HZ, 
XYA. 
H,’’, H,’”, H,/” be the ortho-centers of the trianglas BYZ, XAZ, 
XYH. 
And if 7’; be the homothetic center of the triangles XYZ and i ieee 
T., be the homothetic center of the triangles XYZ and H,” H,” H,”. 
T., be the homothetic center of the triangles XYZand H,’’77,/”H,””. 
Then AT,, BT, CT, concur at the centroid of the triangle X YZ. 
(Nos. 80, 81, 82, 83 are extracted from the work of Dr. Mackay in the Pro- 
ceedings of the Edinburgh Math. Soc.) 
84. If through K parallels be drawn to BC, C/A, AB, they intersect the cor- 
responding altitudes in A,, B,, C,, respectively, which are the vertices of Bro- 
card’s first triangle. BA,, CB,, AC, concur at 2; BC,, CA,, AB, concur at ’, 
and thus the two Brocard points are determined. 
