NEW PROPERTY OF TANGENTS. 339 



Then — and - will be the tangents of HD and Dt. 



Now by Prop. VIII, Sect. I^ Book I, Emerson's Tri- 

 gonometry, 



As radius square — product of two tangents 



Is to radius square, 



So is the sum of the tangents 



To the <angent of their sum), 



...,.,_!_% ,...:i!+r^:'':i±:!i=HK; 



ab a b ab-r* 



therefore a4-b -{ = = the sum of the 



three tangents. 



and 



a'-b-\-ab'' ^ ^ r^a-^r^'b 



, X r* = aft X — ; r-= their continued pro* 



ab—r ab~r^ 



duct. Q. E. D. 



Proposition II. In every obtuse angled plane triangle, 

 the sum of the three tangents of the three angles multiplied 

 by the square of the radius is equal to their continued pro- 

 duct. 



Demonstration. — Let AH, Fig. 4, be an obtuse arc, and in an obtuse 

 and HE, EB the other two. ^"^^""^ '"'^"el«- 



Then BF, ED, and AG are the three tangents. 



Put BF = t and DE = u radius = r, then per trigonome- 



try, as before, r* x — = BT ; 



•" ' r^ — tu 



But - BT = AG = - 4^ X '•'• 

 r^—tu 



Wherefore t -{- ii — -^ — X r* =the sum of the three 

 r^—tu 



tangents, which being reduced 



f + u 



IS = — tu X "n — — ' and mul 



r^—tu 



tu X - —^ X r2 = the product. Q. E. D, 



is = — tux '—-- — ' and multiplied into ;-^ is equal to 

 r^—tu 



r^-tu 



ag IV. On 



