,^i<S PKKSKiX IWJENT A^'l) I'kOOF 1 .\ ( ;i:oM KTK\ . 



and G. Now by drawing (we shall prove it presently) we 

 see that AO^, BO.,, CO3 are concurrent, meeting ni S ; S there- 

 fore in a special way corresponds to C) — it is a sort of avcrayc 

 of the three moves of O. !>o also Ci has its corresponding point 

 K. In these two cases S and K, Ijy virtue of the properties 

 of U and G, acquire a whole set of new symmetries. S of 

 course ( 1 do not pause to prove it) is the Circum-centre, and 

 K ((juite undeservedl}- neglected in elementary text-books ) may 

 be called the Anti-centroid. It has l:)een called the symmedian, 

 as being the point of concurrence of tlie synnnedians ; but it 

 seems to me more convenient to reserve that adjective for the 

 lifics; otherwise why not call the centroid the median? 



These are the two principal pairs ; but it is evident that 

 every point in the plane is given by this triangle its correspond- 

 ing point — its affinity, so to speak. In other words, each 

 triangle polarises the whole plane in its own way point by point, 

 just as each circle polarises it circle by circle, though of course 

 the polarising differs in the two cases. This is a most fimda- 

 mental property. Mr. Johnson, in an admiral)le chapter on 

 the Geometry of the Triangle in his Trigonometry, calls all such 

 pairs of points Anti-centres. I do not like this use of the word 

 centre for points which have nothing really central about them ; 

 I suggest the term Twin-points. Obviously the in-centre is 

 the one point in the plane which is its own twin ; it must be 

 regarded as a double point. It will be found that every point 

 on each of the sides corresponds to the opposite vertex; conse- 

 quently we have to strain the meaning of tunn in the three cases 

 A, B, C: the family becomes rather large. But it is quite as much 

 a strain to say that the whole side I'.C produced to infinity is the 

 anti-centre of A. 



Now this expansion of the triangle through the whole plane, 

 with its scheme of antiparallel^ and the resulting twin-points, 

 opens up a wide range of geometrical ideas. 



Let us hrst take antiparallels by themselves. They will 

 provide us with both a new i)roof and a new view of the Nine- 

 points Circle, which from now I am going to call, from its 

 development, and from its centre being the mid-point of OS, 

 the Ortho-centric Tw^in-point circle. One day, after my class 

 had bisected the sides of a triangle and drawn the consequent 

 triangle of parallels, I told them to draw a transversal anti- 

 parallel to each side. As they had no tracing-paper handy, they 

 hesitated. Thereupon, foreseeing what they would do, I re- 

 minded them of the cyclic property. At once they drew a semi- 

 circle on each side. Then followed a surprised exclamation : 

 " Why, it forms another triangle !" " Just so," I said : " now 

 w^hat triangle is it?" They soon recognised it as the ortho- 

 ■centric. (Fig. 7.) " So, then, looking for antiparallels, you 

 find O again. Now, by the way, what's O to that triangle?" 



