MATHEMATICAL SECTION. 47 
nite line AB to be bisected when a segment equal to one-half the line is 
laid off from either extremity in either direction. 
2. Theorem. The perpendiculars of a plane 
triangle meet in H, and, being prolonged to 
intersect the circumference in A’, B’, C’, the 
segments HA’, HB’ and HC’ are bisected by 
the sides of the triangle. (See Fig. 2.) 
Proof: AB’D = AHD = angle C. 
Notrre.—Here bisection is used in its ordi- 
nary sense. 
HZ is the orthocenter, and the theorem may be otherwise stated as 
follows: The segments of the perpendiculars included between the 
orthocenter and circum-circle are bisected by the sides of the triangle. 
The point of intersection of the medians is the eidocenter,* which 
we call G, and the well-known theorem that the medians are con- 
current and mutually divided into segments of which the greater is 
twice the less may be otherwise stated thus: T'he segments of the 
medians included between the eidocenter and vertices are bisected by 
the sides of the triangle. 
Note.—Here bisection is used in its extended sense. 
Using bisection in this extended sense it is therefore possible to 
unite these propositions into a single general statement, as follows: 
perpendiculars orthocenter 
The segments of the | at between the ROA ction 
away from 
towards the vertices are bisected 
and circumcircle measured | 
by the sides of the triangle. 
Thus in Fig. 3. ye 
oH=alL=}3 HI; and EP=};} EA; 
BH = §M =} HM; EQ =1 EB; 
7H +N = § FN; ER = 3 EC. 
3. Conceive the triangle ABC (Fig. 4) and 4@ 
its cireumcircle O to be the base of an oblique 
cone withinscribed tetrahedron. Let the ver- k 
tex of this cone be so taken that when the whole is projected 
* This term from etdog, form, and xévtpoy, center, has been ‘suggested by 
Mr. Henry Farquhar as being a more accurate derivation than the term 
centroid often used. It is, moreover, analogous to orthocenter, circum- 
center, etc. 
