Properties of Plane Triangles. 31 



Pr. 4-2. ) that the centre of gravity of the triangle is also in OH. 

 ^Whence fmir important points belonging to the triangle are 

 in one line. 



Cor. 3. — The diameter of the circle ati F is half the diameter 

 of the circle circumscribing the triangle ABC. 



For, join LJ, JD, DL. Then this triangle is similar to the 

 triangle ABC and has half its linear dimensions. Hence the 

 diameter of a circle about LJD (viz. the circle ab¥ by Prop. 3.) 

 is half the diameter of that about ABC. 



Prop. V. Plate I. (fig. 2.) 

 Let ABC be a triangle ; e the centre of its inscribed circles ; 

 E, F, G the centres of the circles which touch the sides ex- 

 ternally : and let M, b, s be the external points of contact of 

 the circles with the sides unprolonged. Then the radii Es, 

 Yb, and GM meet in one point. 



For, as is taught in most elementary books, 



As^ + Ci^ + BM^= Sc^ ^ iQ2 ^ MA2; and 



therefore by the converse of a well-known theorem (Bland*, 

 Prop, xxxii. sect. 4.), the perpendiculars from M, s, b meet in 

 the same point d ; and these are identical with the radii of 

 external contact, which radii therefore meet in one point d. 



Prop. VI. (fig. 2.) 



Let the radii of external contact which meet the sides pro- 

 longed, also meet each other in three points (that is say, let 

 GV, EW meet in k, 

 GU, FT meet in /, and 

 EX, Fa meet in in) ; 

 then d is the centre of the circle described through E, F, G. 



Since GE bisects the angle BAV, and its vertical angle 

 CAW ; and in the triangles GVA, GMA the angles GVA, 

 GMA are right angles, the remaining angle VGA (or /zGE) 

 is equal to the remaining angle MGA (or (ZGE). Also, be- 

 cause GkYjd are both perpendicular to AC, they are parallel 

 to one another; and because both E^ and Gd are perpendi- 

 cular to AB, they are parallel to one another. Whence the 

 figure G /■ E ri is a parallelogram ; and having the angle kO d 

 bisected by the diagonal GE, it is a rhombus. 



The like course of reasoning will show that GdFl and 

 dFmE. are also rhonibi. 



The consequence is, that Gc?, <ZE being sides of a rhombus 



• The converse of this proposition in Dland is not necessarily true, ex- 

 cept in case of the triangle; but it is to the triangle we apply it. It is too 

 siui|)le to need formal demonstration here. 



are 



