314 PRKSENTMENT AND PROOF TN GKOM FTRV. 



"Then what can you tell me about the line OP to the circle?" 

 "It is bisected at P." ."Why?" "Perpendicular to the axis." 

 " And what about OD?" " Bisected at D." " Why?" " It's 

 the centre." " As it is the same for the other sides, what can 

 you tell me now about PCJR and DEF?" "A circle will pass 

 through all of them." "And what else will it do?" "Bisect 



all the chord-segments from O." "Especially ?" " OA, 



OB, OC." " Well, then, you have proved the Nine-points 

 Circle. And its size and position?" " Radius half the circum- 

 radius, and centre the mid-point of OS." 



Now the proof given in Mackay's Euclid is based on the 

 following figure (Fig. 3), and everybody can see how uncharac- 

 teristic and repellent it is. And even so, a separate proof 

 depending on a ditTerent principle is required for the size and 

 position of the circle. 



Fig- 3- 



This is as if we set out to ])rove the rationality of man. and 

 started by showing historically that man is " a biped which cooks 

 its food," and so bring it in as a corollary, " hence it should 

 seem that man has some claims to be called rational." 



Our third les.son on the subject was pure Kindergarten. 

 The final result of it is indicated in Fig. 4 ; but I would advise 

 the A^oung teacher to do it with me step by step. " Cut out a 

 large triangle — scalene and acute, because we want the per- 

 pendiculars inside. Pinch the mid-points of the sides. Fold 

 over the joins of the mid-points, and open the triangle again. 

 Now mark ABC and DEF as usual. On the reverse side of 

 the angles ABC mark them PQR. Now fold over A as before: 

 will P fall on BC?" " Yes: the triangles are equal, halves of 

 a parallelogram." "Will P be on the same circle as DEF?" 

 " Yes : the angle P or A is the same as D, opposite in the paral- 

 lelogram." "Then PQR and DEF are on the same circle?" 

 "Yes." "But what are the points PQR?" "Feet of the per- 

 pendiculars." " Fold so as to show this. How do you know 

 it?" "Because when you fold over a triangle you make a 

 symmetrical figure — -a kite ; and its diagonals are at right angles." 



" Then the last three folds were concurrent in the point ?" 



" O." " So it's our old friend again. Now pinch the mid- 

 points of OA, OB, OC— call them UVW. What can you see 

 about the triangle UVW?" " Its sides are parallel to those of 



