PRKSKNTMENT AND PROOF IN GEOMETRY. 



.^17 



P to move along any line AD throngh A. Drop the perpen- 

 diculars PX, PY. and join XY. Do this from more than one 

 position of P. 



Now turn the whole tigure round and it takes the form (2) in 

 Fig. 5. Then if (with tracing paper, and afterwards mentally) 

 the figure (2) he imposed upon (i), it becomes axiomatically 

 evident ia) that A'D' is antii)arallel to AD; (h) that the pro- 

 portion of the perpendiculars (which determines the interior 

 angle) is reversed; and (c) that Y^X^ is antiparallel to XY. 

 These results will be useful in a future proof. 



The method goes further still and brings out a most vital 

 property of the triangle. Consider the following figure 

 (Fig. 6) :— 



Fig. 6. 



Here we have the triangle AF5C turned on its angles so as 

 to form three symmetrical arrow heads. We see at once that 

 the one thing remaining constant is the in-circle with centre I 

 ( the Nine-points and Circum-circle have been unnecessarily 

 drawn). Any other point besides I takes up three new posi- 

 tions on the lie of its antiparallels. I have marked those of O 



