2)22 PRESKXTMENT AN'D PROOF J N C.KO.METKV. 



Simple and obvious as this proof is, I cann<_)t help thinking- 

 it is new; for if it were known, AJr. Johnson would surely not 

 hsve been content with the complex, tliough ingenious^ proof he 

 actually gives — a proof depending on the property that (T being 

 the mid-point of PQ), TD^ = TP= + PD, QG." 



This property can hardly be called fundamental. I did not 

 know it myself, and had to prove it before 1 could proceed, 

 and I should certainly not expect my pupils to remember it.* 



Let me also remark here how the same proof does for both 

 the K-circles and the Twin-point. If you want to kill birds, 

 it is surely good economy to kill two birds with one stone — best 

 of all if you can bring down two whole flocks with one shot. 



Here, of course, is a fresh proof of the Orthocentric Twin- 

 point circle. O and S are twins, therefore DEF, PQR are 

 concyclic, and the centre of the circle bisects OS; and if U be 

 the mid-point of OA. U is on the same circle, either because 

 it is the centre of the circle OQAR, and therefore OUR ^2A = 

 supj)lement of QIMv ; or because FU and EU are parallel to 

 BO and CO, and therefore contain the supplement of A, so that 

 FUE and FDE are supplementary. In fact, there seems to he 

 no limit to the proofs of this wonderful circle. 



The In-circle is. of course, a special case of this family. 

 The twin-points coinciding, the six feet of the perpendiculars 

 coincide two and two. and the circle touches the sides. It is 

 the minimum of the family. 



Perhaps the chief interest of this family lies in the tangency 

 between the Orthocentric and the In-circle (there is no time to 

 discuss the r-circles). A very good way to see the relation l)e- 

 tween the two is to draw tangents to the (Orthocentric parallel 

 to the sides, turning the circle into an In-cirde. ( I'"ig. lo.) I'he 

 joins of corresponding vertices, as the drawing shows, meet 

 where the circles touch. The point II, then, the centre of 

 homology and similitude, may be regarded as generating the 

 whole figure, and we have an infinite series of circles alternately 

 In-circle and Orthocentric to one another. 



Fig. lo. 



* If I criticise Mr. Johnson, it is honoris causa; I wish to give my 

 argument an a fortiori value. His treatment of this topic is the best I 

 know, and I desire to say empliatically that 1 owe it to himself that I am 

 r blc to criticise him at all. 



