112 Mb WALLACE, ON GEOMETRICAL THEOREMS, AND FORMULAE, 



since the angle ABd = ADC is given (figs. 1, 2, 3.)' and also the angle 

 AC(l= ADB; and moreover, because the angles which the lines AB, 

 AC make with a line drawn from S to C may be considered as 

 known, the angles dBC, dCB are known. 



The proposed problem is now transformed to this: 



" Having given all the sides, or else all the angles of two triangles 

 ABC, dBC which have a common base BC, it is required to find the 

 angles which the line Ad joining their vertices makes with the sides." 



This is the geometrical expression of a well known geodetical problem 

 which is more frequently resolved than the other; probably, because its 

 solution is supposed easier. The geometrical property of the figure by 

 which the one problem is converted into the other, namely, the equality 

 of the rectangles AD. Ad and AB.AC is easily remembered, a circum- 

 stance of considerable importance in practical applications of Geometry. 



9. The following Theorem is a deduction from the proposition : 



"If straight lines BD.BE be drawn from B, one of the angles of 

 a triangle ABC, making equal angles ABE, CBD with the sides about 



Fig. 4. 



that angle ; and also straight lines CD, CE, making equal angles BCD, 

 ACE with the sides about another of the angles, and meeting the 

 former lines in D, E, then, straight lines drawn from the remaining 



