PRKSK'.STMENT AN-1) F'KOOF I X GKi»M1:TRV, 



323 



But this tangeucy is by no means unique in the Twin-point 

 family. Indeed, at first I thought it was the usual thing. 

 Several times, drawing at random, I found other Twin-point 

 circles behaving just like the C^rthocentric towards the In-circle 

 —e.g. 



m Fig. II. 



Fig. IT. 



However, on carefully choosing points in all i)ortions of the 

 triangle, I got a very interesting result (Fig. 12). Obviously 

 there is a complex locus of points whose Twin-circles must touch 

 the In-circle. One circle cannot pass continuously into or out 

 of another without touching it. The locus must have double 

 points at the vertices and at I, and apparently passes twice 

 through each side. This becomes evident while we draw the 

 successive circles and imagine those that intervene. To get the 

 full value of this presentment, the student niu ^t have the indus- 

 try to do it for himself. 



Fig. 12. 



Algebraic calculation gives me an equation of the 8th degree 

 for the l<icus ; but its form is not inviting, and I learn more from 

 the Presentment than from this symbolic Proof. 



I will just add my transformation of the proof of the 

 tangency of the Nine-points with the Inscribed circle, as given 

 in Johnson's Trigonometr\-. The only quantitative relation 

 between the two circles given by elementary geometry is that 



