PRi:Si:XTME\T AND I'ROoF IX Cl'J )>.n-:TK^'. 



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ihoug-h in our rudimentary studies we first came upon it as the 

 result of the continence of three easy loci. 



About this series of K-circles Mr. Johnson makes the 

 curious remark that it is unicjue in having its transverse chords 

 parallel to the sides ; but surely he has forgotten the Ortho- 

 centric Twin-point circle, which does not have its centre on SK, 

 but which has this property, and which, indeed, seems to insist 

 on belonging to every family of circles the triangle possesses. 



And now for the last family. As it is not yet named, I 

 am going to call it the Twin-point famil\-. It depends on the 

 fundamental ])roperty of Twin-points. Mr. Johnson puts this 

 property (juite at the end of his chapter as an appendix. Its 

 presentment should come earlier, llie property is that if from 

 each of a pair of twin-points we draw ])erpendiculars to the 

 sides of the triangle, the six feet of the i)er])endiculars are con- 

 cyclic on a circle whose centre is tlie mid-point of the join of 

 the twin-points. Let us prove this at (mce. (Fig. 9.) 



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Fig. 9. 



\Mien we were reversing the triangle bv means of tracing- 

 paper, it l)ecame obvious that any line through a vertex with 

 its series of perpendiculars on the sides and of bases joining the 

 feet of the per])endiculars would ( i ) become antiparallel to its 

 former self, (2) reverse the proportiim of the perpendiculars, 

 and (3) make the bases anti])arallel to what they were. In this 

 figure (9) let any ])oint P be taken and BQ, CQ be drawn anti- 

 parallel to F>P. CP; we must now first show that AQ will also 

 be antiparallel to AP. It is obvious that QH and QG reverse 

 the })roportion of PE and PD ; similarly, QK and QG reverse 

 the proportion of PF and PD ; combining these proportions, we 

 see that OH and OK reverse the proportions of PE and PF — 

 i.e., Q is on the antiparallel to .\P ; therefore, if P be any point, 

 the antiparallels of PA, PB, PC through the vertices are con- 

 current. Then, since the bases HG. etc., are antiparallel to 

 their former selves (DE, etc.), DGEH is cyclic; so is EHKF. 

 The centres of these two circles are v,-here the symmetrical 

 bisectors of DG, EH, KF meet: but these, running midway 

 l)etween the perpendiculars, are obviously concurrent at the 

 mid-point of PQ. Hence the six feet of the perpendiculars of 

 twin-points are concyclic ; and the centre of the circle lies mid- 

 way between the two points. 



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