388 THE NINE-POINTS CIRCLE. 



This gives the proportion D/.DP=DX- 



But DL.DP=r:DX= 



i.e., I coincides with L 

 .• . in this circle DP subtends the angle 2 LIX or (C — B) at the 

 circumference, 



But in the triangle it is known (or can easily be proved) 

 that DEP, DFP, DQP, DRP, DTP, DUP, DVP, are all = 

 (C-B). 



. • . DEF, PQR, TUV are all on one circle which touches the 

 in-circle, and which (since the reasoning applies equally) touches 

 the ^-circles also. Thus in one proposition we have proved the 

 existence and all the chief properties of the Link or Nine-points 

 circle, including Feuerbach's theorem. 



This proof may seem clumsy and over-detailed, but it has 

 the merit of showing how the asymmetries of the triangle keep 

 in touch with the symmetrical stage. As the angle A moves, 

 its departure from symmetry is measured lineally by the segment 

 DP, and angularly by B'^C. The proof, then, shows how all 

 the eleven other points unite their angular variation with the 

 linear variation of D and P. 



For reference, I append the seven little proofs assumed 

 above : — • 



(i) and (2) D and P. 



(3) DFP=DFA— PFAr=i8o°— A— 2B=C— B. 



(4) DEP=DEC— PEC=A— (i8o°— 2C)=C— B. 



(5) DRP=BRP— BRD=r:C— B. 



(6) DQPr^DQC— PQC=C— B. 



(7) DUP^BUP— BUD=2C— (i8o°— A)=C— B. 



(8) DVP=DVC— PVC=:l80°— A— 2Br=:C— B. 



(9) DTP. 



If OP is produced to H so that PH^OP, and OD is pro- 

 duced to K so that DK=OD, then H and K are on the circum- 

 circle ; and since HK ] | DP, the angle at H is a right angle, and 

 AK is a diameter. 



.-. DTP=SAP=C— B. 



Genesis of the proportion DL.DP=:DX'^. 



Hitherto I have been content with asserting the well-known 

 relation in the triangle, DL.DP=DX-. 



The following consideration will show how important for 

 our purpose it is to study it a little more deeply. If on a 

 straight line we have a double mean-proportion range with a 

 common segment dp, so that dp.dl=dx-=d.rl and dp.dV-=^dxl 

 =dx% and if circles be drawn as in the accompanying diagram 

 (Fig. 5), it is obvious that a circle through d, p, touching the 

 t-circle will also touch the 4-circle ; and one touching the 

 4-circle will also touch the /,- circle. In one of these two circles 

 dp will obviously subtend an angle=2 lix, and in the other an 

 2ingle:=2l^it,Xb (or their supplements). If, moreover, HtXt is 



