THE NINE-POINTS CIRCLE. 



389 



the complement of lix (i.e., if its double is the supplement"), it 

 "becomes clear that dp will subtend the same angle towards the 

 same parts in both circles: that is, the two circles are identical. 

 In this case, then, we have put four circles in such a position 

 that one circle can be drawn touching all four. We are thus 

 very near to Feuerbach's theorem, and it becomes interesting to 

 see how the four circles can be thus placed. 



The natural home of the mean proportional is the circle. If 

 a point H moves towards a circle, and in each position a circle is 



Fig. 5. 



struck from it as centre through the two points of tangency, its 

 whole pencil of chords is divided by itself and the four arcs in 

 double mean proportion. A similar pencil is formed when H 

 has entered the circle and strikes another circle on its symmetrical 

 chord as diameter. . These two pencils are, however, limited by 

 the breadth of the original circle. But when H is on the cir- 

 cumference, a very surprising thing happens. Instead of the 



