MR TALBOT ON MALFATTl's PROBLEM. 129 



178), a geometrical construction of singular simplicity, but entirely unaccompanied 

 with proof. His solution is as follows : — 



Let ABC be the triangle in which it is required to inscribe three circles touch- 

 ing each other, and each of them touching two sides of the triangle. Bisect the 

 angles A, B, C, by three lines AO, BO, CO, which will meet in the point 0. In 

 the three triangles AOB, BOC, COA, inscribe three circles, touching the sides of 

 the given triangle in the points D, E, F, which letters will serve to denote those 

 circles respectively. From the point of contact D, draw a line DG, touching the 

 circle E on its inner side. Similarly from the point of contact E, draw a line EG 

 touching the circle D, on its inner side. And let DG, EG, intersect in G. Then 

 we have a trapezium BDEG, and Steiner affirms, first, that a circle can 

 always be inscribed in this trapezium ; and, secondly, that the circle so inscribed 

 will be one of the three required circles. 



But no doubt a difficulty will be observed immediately. Steiner directs that 

 the line DG shall be tangent to the circle E ; and plainly there is no reason why 

 the circle E should be selected rather than the circle F. But the reply to this 

 is, according to Steiner, that if the line DG touches the circle E, it will also 

 necessarily touch the circle F. 



But of this most remarkable theorem he gives no demonstration whatever, 

 although there is assuredly no theorem in the whole of geometry which has less 

 claim to be considered as an axiom. Moreover, he affirms that the same line, 

 DG, touches two of the required circles of the problem, at the point where they 

 touch each other. This being admitted, the construction of the problem follows 

 at once, as it is only requisite to describe a circle touching AB, BC, two sides 

 of the given triangle, and also the known line DG, and this circle will be one of 

 the three circles required, the others being found with equal facility. 



Steiner's solution, therefore, would have left nothing more to be wished for, 

 if it had been accompanied with a demonstration. But of such his memoir con- 

 tains not a single syllable. He says, indeed (page 178), that this solution of a 

 difficult problem shows "the fruitfulness of the preceding theory;" but the 

 critical researches of subsequent inquirers have not failed to discover the singular 

 circumstance, that there is no connection whatever between this solution of 

 Malfatti's problem and the theories set forth in the preceding part of Steiner's 

 memoir. 



The great simplicity and elegance of this solution discovered by Steiner 

 rendered a demonstration of it very desirable, which was at length accomplished 

 by Zornow of Konigsberg, in Crelle's Annals for 1833 (vol. x. p. 300). The 

 demonstration of Zornow is remarkably elegant, but it chiefly depends upon 

 some very dexterous algebraic transformations, in the course of which, how- 

 ever, all perception of a geometric proof of the construction necessarily dis- 

 appears. 



