MATHEMATICAL SECTION. 49 
joining the circumcenter to the orthocenter, and are points of 
bisection, in the ordinary and extended sense, in such manner that 
i= FO = PO — FF" A FO. 
(c.) The segments of the perpendiculars HA, HB, HC, and HA’, 
HB’, HC’ are all bisected by each of these circles. In the case of 
the first circle F, the segments are bisected in the ordinary sense, 
and since the segments HA’, HB’, and HC’ are also bisected by 
the sides of the triangle this circle passes through the feet of the per- 
pendiculars. 
The points of bisection on the perpendiculars determined by the 
circle F are points of bisection in the ordinary sense. 
The points of bisection on the perpendiculars determined by the 
7 
circle | are points of bisection in the extended sense, and in 
such Wise that the segments cut off from A and A’, B and B’, Cand 
towards 
away from the vertices. 
C’ are measured from the orthocenter | 
(d.) Every line drawn from # to the circumcircle is bisected by 
the three circles F, F’, and #’”’. His therefore a direct center of 
similitude common to the circles O, F, and F” and an inverse 
center of similitude common to all four circles. 
4. The eidocenter is collinear with the circumcenter, orthocenter, 
and Feuerbach center*; for from a known theorem we have 
HB = 2 OM, and therefore HO must divide BM into segments of 
which the greater is twice the less, 7. e., it must pass through the 
eidocenter. 
5. Again conceive the triangle ABC (Fig 5) as the base of an 
oblique cone, etc, as in section 3, except that its vertex is to be con- 
ceived as perpendicularly over the eidocenter instead of over the 
orthocenter, and the whole projected as before. In this case KA, 
EB, EC,and EA’, EB’,EC’ are projections of the elements of the 
cone and EO, coincident with HO from section 4, the projection 
of the axis. 
Let the axis be bisected as before by three planes parallel to the 
base: one midway between apex and base, another below the base, 
and a third above the apex, and the sections so formed projected 
upon the plane of the base. 
* The center of Feuerbach’s circle may be so called for brevity. 
