THE NINE-POINTS CIRCLE. 



385 



circle's nine points with its centre, and also two of the r-scribed 

 circles. The figure now shows the Link circle (QDR) in its true 

 light, classified with the sides as the fourth tangent to the two 

 remaining scribed circles. 



Throughout this movement one cannot but notice the per- 

 petually recurring proportion DL.DP = DX-. It is more than 

 a leit-motif; it reminds us of the famous dactyl and spondee 

 rhythm on which the slow movement of Beethoven's Seventh 

 Symphony is built. The whole scheme might be called a 

 geometrical symphony of mean proportionals. 



It was this thought that led me to a new proof of Feuer- 

 bach's theorem. It seemed to me that an analysis of this pro- 



Fig. I. 



portion separately in the movement of the lines and of the 

 circles might explain the power which the Nine-points Circle has 

 of linking together into harmony all the asymmetries of the 

 triangle and its associated circles. 



Analysis of the mean proportion dl.dp--=dx- : — If a line- 

 segment of constant length iv be moved upon a fixed line-segment 

 dp, there are only two positions in which the mean proportion 

 can hold, and these are simply reversed readings of each other. 

 This is made obvious by revolving the segment Ix to the position 

 xl^ (as in Fig. 2). The proportion dl:dx:: dx: dp can be trans- 

 formed into dl'.lx:: xP- : P-p, which, of course, reads both v/ays. 

 The algebraic proof that no other position is possible is very 

 simple. It can be seen with equal simplicity in this geometric 

 figure. (Fig. 2) :— 



• Fig. 2. 



