C 120 3 
VIII. On a new Property of the Tangents of the three Angles of a 
Plane Triangle. By Mr. William Garrard, Quarter Master of 
Instruction at the Royal Naval Asylum at Greenwich. Com- 
municated by the Astronomer Royal. 
Read February 11, 1808. 
Proposition I. In every acute angled plane triangle, the 
sum of the three tangents of the three angles multiplied by 
the square of the radius, is equal to the continued product of 
the tangents. 
rn 
Demonstration. — Let AH, HI, and IB 
be the arches to represent the given 
angles ; and AG, HK, and BT be their & 
tangents, put r the radius, AG = a, and 
BT = b, 
y.'it 
Then - and ^ will be the tangents of 
HD and DI. 
Now by Prop. VIII. Sect. I. Book I. 
Emerson’s Trigonometry, 
As radius square — product of two tangents 
Is to radius square. 
So is the sum of the tangents 
To the tangent of their sum. 
G- 
A /I 
Jk. 
yxN. 
a 
7*4 
• f *— - t • 
ab 
r -^±4 = HK ; 
a 1 b ab — r a 
therefore a -f- b -f- r = a f +aZ l 1 = the sum of the three 
1 1 ab — r x ab — r a 
tangents, 
j a x b -1- ab 1 
a" d ~zr=TF * r 
O. E. D. 
ab x = their continued product. 
