320 PRKSENTMENT AND I'ROf)K IN GEOMKTKV. 



ceiitroid circle : the utiier name is .derived from a subordinate 

 property. 



FiR. 8. 



'I'he Anti-centroid circle, as we shall see presently, is the 

 minimum member of an interesting group of circles, each cutting 

 the triangle so as to have three transverse chords parallel to 

 the sides, and three antiparallel to them. Let us now intro- 

 diice a new member of the family. Through K draw three 

 parallels to the sides ; these, of course, form three parallelograms 

 with the sides (Fig. 8) ; and KA, KB, KC bisect the other dia- 

 gonals. But the transversals which KA,KB,KC bisect are the anti- 

 parallels ; and parallels with antiparallels form cyclic quadri- 

 laterals ; thus the six ends of the parallels through K are four 

 and four concyclic in three sets. The centres of these circles 

 are where the symmetrical bi.sectors of the antiparallel chords 

 meet. But these bisectors are parallel to SA. SB, SC (which 

 are respectively perpendicular to the antiparallels), and as they 

 start from the midpoints of KA, KB, KC, they must be con- 

 current at the midpoint of SK. The three circles, therefore, 

 are the same, and this is a simple proof of what is called the 

 Lemoine circle. 



It is easily seen, as Johnson points out — and it can be proved 

 by quite similar reasoning — that if KA, KB, KC are divided in 

 any other ratio, a set of circles all centred on SK cut the sides 

 of the triangle with parallel and antiparallel transversals. If 

 again J be the moving paint, then, when the ratio KJ : JA is 

 o, we have the minimum, the anti-centroid circle; when i, the 

 Lemoine ; Avhen o© , the Circum-circle, because there the 

 parallels merge into the sides and the antiparallels vanish into 

 the vertices (i.e., the tangents to the Circum-circle at the 

 vertices are antiparallel to the sides). The whole family is 

 called the Tucker circles ; I do not see why we should not call 

 them the K-circles, and the Lemoine the mid-K-circle. The 

 other names seem to give Messrs, Lemoine and Tucker some- 

 thing too much of precedence in Geometry. 



It will be observed that here again the Circum-circle occurs 

 not as a primary, but as a derived adjunct to the triangle, 



