THE 



SOUTH AFRICAN JOURNAL 

 OF SCIENCE, 



COMPRISING THE REPORT OF THE 



SOUTH AFRICAN ASSOCIATION 



FOR THE 



ADVANCEMENT OF SCIENCE. <^ 



(1919, KINGWILLIAMSTOWN.) 

 Vol. XVI. APRIL-JULY, 1920- No. 5. 



THE NINE-POINTS CIRCLE— A NOTE ON 

 FEUERBACH'S THEOREM. 



By THE Rev. F. C. Kolbe, D.D., D.Lit. 



With 5 Text-figures. 



Read July 9, 1919. 



[In the following paper I adopt the usual notation: ABC 

 for the triangle ; DEF for the feet of the medians ; LMN for 

 the feet of the angle-bisectors ; PQR for the projections of 

 ABC on the sides ; I, etc., for the centres of the scribed circles ; 

 XYZ, etc., for the projections of I, etc., on the sides. Points 

 analogous to DXLP I call d.vlp.] 



Three years ago I read a paper on the teaching of Geometry, 

 illustrating my thesis by a number of proofs of properties of 

 associated circles of the triangle. My thesis was that if pre- 

 liminaries are rightly presented proofs can generally be expressed 

 with great simplicity. It struck me then that Feuerbach's 

 Theorem — viz., that the Nine-points Circle touches all the four 

 scribed circles — ought to be capable of such a proof. The fact 

 that nobody, as far as I knew, had found one seemed to 

 me to be a hint that we had not struck on the right 

 way of presenting it. A subsequent study brought me 

 three or four such proofs, which I now bring forward. The 

 last form of the proof is of the utmost simplicity; and if I were 

 talking to geometricians only, my paper need not be more than 

 ten or twelve lines in length. But I am still illustrating a 



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