PHILOSOPHICAL SOCIETY OP WASHINGTON. 55 



color-blindness, the errors committed will determine not only the 

 existence of the defect, but also the particular colors which the 

 •eye is incapable of distinguishing. 



DISCUSSION OP A GEOMETRICAL PROBLEM, WITH BIBLIOGRAPHICAL 

 NOTES. BY MARCUS BAKER, U. S. COAST SURVEY, WASHINGTON, 

 D. C. 



The problem here discussed, and of which several solutions are 

 given, is the following : — 



In a right-angled triangle there are given the bisectors of the 

 acute angles : required to determine the triangle. 



This problem, like most problems in triangles in which the 

 bisectors of the angles enter as a part of the data, cannot be solved 

 t»y the elements of geometry, i. e. by the use of the circle and 

 right line only. We shall give, first, trigonometrical solutions ; 

 second, algebraical solutions ; third, constructions ; and fourth, 

 bibliographical notes. 



PIRST SOLUTION. 



Let a and ^ be the bisectors of the angles A and B respect- 

 ively : then we have 



AB sin A=)3 cos (45° -^ A) and A B cos A. = a cos A A ; 

 whence by dividing, remembering that 



cos (45°—^ A) _ 1 + tan ^ A 

 cos I A ~ ^3 



and since 



tan A=— ^(1 -ftan^A) : . . . (1) 



av/2 ^ ^ 



^ , 2 tan i A 



we obtain by reduction 



tan«iA + tan»iA-f Q^8 — l)taniA — 1=0 ... (2) 



from which equation we may find tan |- A. 



We may, however, obtain Eq. (1) directly from a construction 

 as follows : — 



Prolong A C to E' making C E' = C E, and from E' draw 

 E' G perpendicular to A E' : from E draw E F perpendicular to 

 A E, meeting E' G in F ; and from C draw C G parallel to E F. 

 Now the triangle C E' G is equal to the triangle ACE; hence 



