136 



MR TALBOT ON MALFATTI S PROBLEM. 



and the more general theorem of which they are particular cases, are original ; at 

 least I am not aware of their having been published elsewhere. They supply the 

 link that was missing in Plucker's investigation, and singularly facilitate the 

 demonstration of Steiner's elegant construction. 



By the help of the preceding Lemmas, we can show the truth of that con- 

 struction in the following manner : — 



Malfatti's Problem. 



It is required to inscribe in the triangle ABC three circles touching each other. 



and each of them touching two sides 

 of the triangle. 



Solution. — Suppose the thing done, 

 and the three circles A', B', C, found, 

 it is plain that their three internal 

 common tangents meet in a point N, 

 and bisect the external tangents GH, 

 IJ, KL. 



Produce the lines EN, FN beyond 

 the point N until they meet the base 

 BC in two points Q, R. Then if in 

 the triangle QRN so formed, we in- 

 scribe a circle which may be * called a, it will touch the base BC in the point D 

 (by Lemma 3). 



By an exactly similar process we obtain a circle (3, touching DN, EN produced, 

 and the side AC in F ; and a circle 7 touching DN, FN produced, and the side 

 AB in E. 



A 



Fig. 10. 



* I have called it a, because it stands opposite to the angle A of the original triangle, 

 larly for the names of j3 and y. I have called a, j3, y the secondary circles. 



Simi- 



