MR TALBOT ON MALFATTI S PROBLEM. 



131 



Fig. 1. 



can be no doubt that an easier proof will be found of it." And he adds a wish 

 for "a simple geometrical proof." 



In the demonstration which I now submit, I shall follow Plucker's geometrical 

 proof in a general way, up to the point where he breaks 

 away from geometry into the regions of analysis, and then 

 give a proof, by geometry alone, of the remaining portion 

 of the investigation. 



Lemma 1. — If a circle is inscribed in a triangle, the 

 difference of the sides equals the difference of the seg- 

 ments of the base. 



This is evident. D, E, F, being the points of contact, 

 AB — AC = BE — CF = BD — DC. 



Lemma 2. — Let two circles touch each other at 0, and let BFGC be their 

 common external tangent, and AOD their 

 common internal tangent. Let A be any 

 point in the tangent AOD, and let AEB, 

 AHC, be drawn touching the circles ; then 

 if a circle be inscribed in the triangle ABC, 

 it will touch the base BC at D. 



Demonstration. — We have manifestly the 

 equal tangents AE = AH, DF = DG, BE 

 = BF, and CG = CH. Therefore AB - 

 AC = BE - HC = BF - GC = BD - DC. 

 Therefore by Lemma 1 the inscribed circle touches the base in D. 



Lemma 3. — This is only another case of Lemma 2, when the point A is taken 

 so near to that the tangents AE, 

 AH, diverge from the base BC, but 

 their prolongations AB, AC, intersect 

 the base at B, C. 



In this case also we have the equal 

 tangents AE = AH, DF = DG, BE = 

 BF, and CG = CH. Therefore AB - 

 AC = BE - HC = BF - GC = BD - 

 DC. 



Lemma 3 is the case which occurs 

 in the solution of Malfatti's problem, but as the same demonstration applies to 

 Lemma 2, I have given both of them. 



Lemma 4. — If tangents of equal length are drawn to acircle, the locus of their 

 extremities is a circle concentric to the first. 



Lemma 5.— Let there be two circles A, B, and let RS and its equal RS be their 

 two common internal tangents; then if OP, OQ are two tangents drawn from 



VOL. XXIV. PART I. 



2o 



