Sept., 1887.] ELLIOTT SOCIETY. ' 181 



angles by using second term as altitude, which alters not the proportionality, 

 and get AIxAE : AIxED :: AIxED : KIxED. But by Prop. VI, each of 

 these means is equal to the area ABC ,and by Prop. V the last term KIxED 

 equals BExBI, substituting these quantities in last proportion, we shall have 

 rectangle AIxAE : area ABC:: area ABC : rectangle BExBI, which proves 

 the proposition. 



The above porportion expressed in numbers leads to the usual arithmetical 

 rule. 



10. The first term of the final proposition given above, is the rectangle Alx 

 AE, AI being the half -perimeter, and AE the difference between that half- pe- 

 rimeter and the base BC, or side opposite to vertex A, of the angle which had 

 been bisected by the line AK. The last term is the rectangle BExBI ; BE 

 being the difference between the half -perimeter and side AC, one of the sides 

 about the angle A, BI being the difference between the half -perimeter and side 

 AB, the other side about the angle A. Hence it will be evident that if we bi- 

 sect angle B by a line which we may call BDK', the fii-st term of the final propor- 

 tion will be the rectangle AIxBE, and the last term will be AExBI ; also 

 that if we bisect angle C by a line, which we may call CDK" that the first and 

 last terms of the final proportion will be AlxBI and AExEB. Therefore 

 three pairs of rectangles can be found, between the members of each of which 

 the area ABC is a mean proportional. 



The points D, K, K'. and K" are well known as the centres of the inscribed 

 and of the three escribed circles, but as no property of the circle is used in the 

 demonstration, it is not necessary to draw those circles. 



11. The preceding demonstration may be regarded as complete, as we have 

 only assumed that the reader is acquainted with the simplest theorems relating 

 to equality of triangles, to ratios and proportions, and to the proportionality of 

 similar triangles ; if introduced into a course, the above propositions could be 

 placed among the propositions of Legendre's fourth Book. 



12. Although none of the circles above mantioned were required in the pre- 

 ceding demonstrations, we add the following known propositions relating to 

 their radii, as they are closely connected with our subject. 



Prop. VIII. The area of a triangle is equal to that of the rectangle formed 

 by the radius of the inscribed circle and the half -perimeter. 



This is evident from Prop. VI, which shows that the area is equal to AI X 

 ED ; for AI is the half -perimeter, and DE is one of the three equal perpendic- 

 ulars DE, DH, and DF, and is therefore the radius of the inscribed circle. 



Prop. IX. The area of a triangle is also equal to that of the rectangle form- 

 ed by the radius of one of the escribed circles, and the line which is the differ- 

 ence between the half -perimeter and that side which is tangent to that escribed 

 circle. 



For by the diagram, AE : DE:: AI : KI ; whence the rectangle DExAI 

 equals rectangle KIxAE. By last proposition the first rectangle DEXAl 

 equals area of triangle. In the second rectangle, KI is, by Prop, II. one of 

 the three equal external perpendiculars, KI, KL, and KM, and is therefore the 



