390 THE NINE-POINTS CIRCLE. 



supply of mean proportions vanishing, it is infinitely multiplied. 

 If, now, a circle be described from H as centre through B, C, 

 the ends of any chord symmetrical to H, then the two circles and 

 the straight line BC divide in double mean proportion every 

 chord through H ; and this pencil extends to infinity on both 

 sides. So that we have an infinite number of infinitely extended 

 pencils. Since the same thing happens with H^, the other end of 

 the H diameter, it follows that both H and H^ can choose the 

 same chord BC, and thus concentrate their mean proportionals. 

 Both sets can be projected on BC, together with a common seg- 

 ment, and we shall have the quadruple mean proportion we are 

 looking for. Looking again at thar original circle and its two 

 auxiliaries struck from H and H^ through B, C, we find (from 

 the angles of the arcs on BC) that these two auxiliary circles 

 are the locus on which lie all the in- and e-scribed centres of 

 ABC, as A moves on the original circle. Since in each such 

 position of A, HA, and H^A are the bisectors of the angles at A, 

 it follows that I, Ia>lb,le, are the four points where HA and H^A 

 meet these auxiliary circles. Owing to this property, the whole 

 scheme of the triangle, with its five principal associated circles, 

 can be drawn zvithoiit any measurement dt all. Moreover, we see 

 now that the quadruple mean-proportion range projected on BC 

 has the relation, not only of the common seginent DP, but also 

 of the angle Ll^Xt being complementary to LIX — thus making 

 it obvious that a circle through D and P touching one of the 

 scribed circles touches them all. 



We are now able to give an almost axiomatic proof of the 

 existence of the Nine-points Circle along with the Feuerbach 

 property. 



Final Proof. 



From the properties of the circum-circle in relation to the 

 triangle ABC we have shown that there is a circle passing through 

 D, P, which touches all four scribed circles. 



For the same reason, there are circles passing through E, Q, 

 and through F, R, touching the same four circles. 



But only one circle can touch three given circles in the same 

 sense. 



Therefore these three circles are one and the same. That 

 is, there is one circle passing through DEF and PQR. and touch- 

 ing all the in- and (^-circles. 



It is an exceedingly simple matter to add the other properties 

 of this Nine-points Circle. 



I think it will be conceded that this genetic way of regarding 

 the question is more fruitful and suggestive than isolated proofs. 

 And I think I may claim to have once more made good my thesis 

 that in geometry suitable preliminary presentment almost super- 

 sedes proof by making intuition possible. 



