MATHEMATICAL SECTION. 51 
The points of bisection on the medians determined by the circles 
wt ry 
| yf are points of bisection in the extended sense, and in such 
manner that the segments cut off from A and A’, B and B’, C and 
towards 
away from 
(d.) Every line drawn from £ to the circumcircle is bisected by 
the three circles F’”’, F’”’, F'; E is therefore a direct center of simili- 
tude common to three circles O, F”, F’” and an inverse center of 
similitude common to all four circles. 
OC’ are measured from the eidocenter i the vertices. 
6. If we now compare figures 4 and 6 and consider the three 
circles in each which resulted from projection, we find that circle F, 
and only circle F, is identical in the two figures. 
This identity appears from the fact that their diameters are equal, 
each equaling half the diameter of the circumcircle, and their cen- 
ters coincident. The coincidence of their centers appears from 
drawing the projections of the axes of the cones side by side, thus: 
O F Lik 
_——— From Fig. 4. 
|———— - —- ve From Fig. 6. 
O E F 
From Fig. 4 we find that this circle, Feuerbach’s, passes through 
6 noteworthy points, being 2 points on each perpendicular; and 
from Fig. 6 we find that the same circle passes through 6 other 
noteworthy points, being 2 on each median, or in all Feuerbach’s 
circle passes through 12 noteworthy points. 
7. It is apparent from the foregoing that if we should not bisect 
the axis of the cone but should cut it in any ratio (Fig. 4) by a 
plane parallel to the base then the projection of the section would 
bea circle cutting the perpendiculars in that ratio and its center F, 
would divide HO in that ratio. The medians would not be divided 
in that ratio, but a point in HO exists (ca!l it E,) through which, 
if lines be drawn from the vertices to the opposite sides, these lines 
would be divided in the given ratio. 
The four points O, En, /,, H would in this case, as before, be 
four harmonic points. 
8. Since Feuerbach’s circle bisects the sides of the triangle it is 
the circumcircle of a triangle similar to the original and of half its 
