

REMARKS UPON CLIFFORD'S PROOF OF MIQUEL'S 



THEOREM. 



By F. H. Loud. 



The name of Auguste Miquel, on the tongues or pens of 

 geometers of the present day, occurs most frequently in con- 

 nection with the remarkable theorem* which forms the con- 

 cluding proposition of the following statement. 



Given five lines in a plane, they form ten triangles, whose 

 oircumcircles meet by fours in five points, and these points 

 lie on a circle. 



For convenience of statement, I have combined with 

 Miquel's theorem proper an antecedent truth upon which it is 

 based, relating to four lines. On examination it will be per- 

 ceived tliat if we would build up the theorem from its ele- 

 ments we must begin with two lines, in the following fashion: 



Given two lines, they have one point of intersection, F.^. 



Given three lines, we may leave out one at a time, and 

 thus form three pairs. Each pair has its point of intersec- 

 tion, P2, and the three points lie on a circle Cg. 



Given four lines, leaving out one at a time forms four 

 sets of three, each with its circumcircle C,, and these four 

 circles meet in a point P ^. 



Given five lines, they form in the same way five sets of 

 four, each determining one point P4, and these five points lie 

 on a circle Oj. 



The interest of Miquel's theorem was much increased 

 when it was shown by W. K. Cliff ord.f and later, (though, it 

 seems, independently) by S. Kantor,J that the series of 

 propositions thus begun continues true in an indefinite pro- 

 longation, defining, for every even number, 2?i, of arbitrarily 



* Liouville'.s Journal, vol. x, p. 349. 



f'Syntlietic Proof of Miciuol's Thoorem," Mathematical Papers, Tp.^. 

 X " Ueberdeu Zu.sammculiang von n beliebigcn Goraden in derEbcno," Sitzungs- 

 bericbte, Wien, 1«78, p. 789. 



