PARTICULARLY APPLICABLE TO SOME GEODETICAL PROBLEMS. 113 



angle A to D and E shall make equal angles BAD, CAE with the 

 sides about the angle*." 



The well known proposition, that the lines which bisect the three 

 angles of a triangle meet in the same point is a particular case of this 

 Theorem. 



10. The Trigonometrical solution of the Geodetical Problem of 

 Art. 1. which is deducible from the construction here given, is suffi- 

 ciently obvious : I shall therefore, without at present entering into it, inves- 

 tigate another theorem which comprehends the former and various others. 



Let the sides of any triangle be a, b, c, (fig. 5, 6). 

 The opposite angles A, B, C 



Fig. 6. 



Fig. 5. 



Let straight lines be drawn from any point D to the angles of the 

 triangle, and put 



AD^x, BD = y, CD=%. 



AAA 

 Also the angles yz = a, a;a = /3, xy = y. 



* I owe this elegant proposition to T. Galloway, Esq. F.R.S. to whom it occurred 

 •when considering the Theorem. 



Vol. VI. Part I. P 



