88 
SoME CrIRcLES CoNNECTED WITH THE TRIANGLE. 
By Rost. J.,ALEY. 
In my study of the geometry of the triangle I have frequently felt the need 
of an available collection of the circles connected with it. So far as I know, no 
such collection is extant. The following list, which is by no means complete, is 
offered as the beginning of what it is hoped may grow into an exhaustive 
collection. 
1. Cireumcirele.—The circle that passes through the vertices A, B, C of the 
triangle. Centre at M, the point of concurrence of perpendiculars erected at the 
mid-points of the sides. R, the radius = “4 
2. Incirele.—The circle which is tangent internally to the three sides of the 
triangle. Centre at I, the point of concurrence of the three internal bisectors of 
the angles of the triangle. r, the radius = Ss 
3. Erxeireles. The three circles which are tangent externally to one side and 
internally to two sides of the triangle. Centres are I,, I,, I,, the points of con- 
currence of the external bisectors of the angles with the internal bisectors of A, 
: eo A A A 
B, C, respectively. The radii are r; —=———, r, = —— ‘and r, = 
s—a s—hb 
s—c 
4, Nine Points Circle.—The circle which passes through the midpoints of 
the sides of the triangle, the feet of the perpendiculars, and the midpoints of the 
parts of the altitude between the orthocentre and the vertices. Centre is at F, 
the midpoint of IM. The radius is} R. It is tangent to the incircle and to each 
of the excircles. 
5. Brocard Circle.—The circle whose diameter is the line joining the circum- 
centre M, to the symmedian point K. It passes through the two Brocard points 
Q,Q, and through the vertices of Brocard’s first and second triangles, Centre at 
midpoint of MK. 
6. Cosine Circle.—The circle which passes through the six points of inter- 
section of antiparallels through K with the sides. Centre is at K (Symmedian 
point). 
7. Ex-Cosine Circles.—The three circles which have K,, K,, K,; (ex-sym- 
median points) for centres, and which pass through B, C; C, A; and A, B, re- 
spectively. 
8. The Lemoine Circle.—The circle which passes through the six intersections 
of parallels through K with the sides of the triangle. The centre is at the mid- 
