THE NINE-POINTS CIRCLE. 



387 



Now DWP=C— B. 



. • . min-=2 (C — B). 

 and mix:=C — B. 



.• . Ix subtends at i the same angle as LX does at L 



i.e., /.t— LX, 

 and as x and X are obviously on the same side of the mid-point 

 of DP, Ix coincides with LX, 



. • . the moving circle coincides with the in-circle. Ergo, 



As precisely similar reasoning applies to the (^-circles, this 

 demonstrates Feuerbach's theorem. 



Remarks. — If any ultra-conservative objects to sliding a 

 circle (though every boy can trundle a hoop), it is quite easy to 

 draw the circle in position, since the altitude of the centre and the 

 two radii are known. 



Although it is only a petty point in pedagogy, yet I may 

 add that this line of proof gives an easy way of drawing the 

 whole scheme. By drawing first two touching circles, the tri- 

 angle and all the rest can be built up fairly accurately without 

 any measurements, using only ungraduated ruler and compasses, 

 and applying the set-sc|uare four times. Indeed, on occasion I 

 have, though no great draughtsman myself, chalked on a black- 

 board for my class the whole figure, triangle, and five circles, 

 quite freehand — a thing I should find impossible if I drew the 

 triangle first. 



I should not have mentioned this, except that it suggested 

 to me a new and comprehensive proof. If, I thought, the figure 

 can be built up from two circles and one proportion, why not 

 also a demonstration? 



Simultaneous proof of all the properties of the Link Circle. 

 — Given a triangle and its in-circle, with PQR, DEF marked, and 

 also O and TUV the mid-points of OA, OB, OC. 



Fig. 4. 

 Draw a circle passing through D and P, and touching the 

 in-circle at W. 



