of Edinburgh, Session 1884-85. 337 
With the usual notation, let the equations to the sides of the 
triangle ABC be 
y = m 1 x + c 1 , y = m 2 x + c 2 and y = m 3 x + c 3 . 
Then, since twice the area of any triangle ABC is 
a 2 
we have 
2AABJA = 
cot B + cot C 
(BA ) 2 
( C 3 
_ \ 3 
2 3 cot B 9 + cot Bo m 0 - m c 
with similar expressions for the triangles BBgBj and CB^ . 
But ABC = AB 3 B 2 + CB x B 2 - BB 1 B 3 , 
... 2 A ABC = • 
m 2 - m 3 va 3 - m 1 m 1 - m 2 
Again, by reference to the triangles formed by OX and the sides of 
ABC produced, since 
(OA 2 -OAo) 2 (m 2 c»-nioC 9 ) 2 
2 A AAoAo = - 7-r- — rr ~ = 2 ~ 3 ; 1 
1 6 cot A 2 + cot A 3 m 2 m 3 ( m 2 — m 3 ) 
with similar expressions for the triangles BAgAj and CAjAg, w 
have also 
2AABC 
= (m 2 c 3 -m 3 c 2 ) 2 t ^ (m Y c 2 - m 2 ctf . 
m 2 m 3 (m 2 - w 3 ) m 3 m 1 (m 3 — m x ) m 1 m, 2 {ni 1 — m 2 ) 
And each of these expressions for 2AABC may be shown to be 
equal to 
{ c l (m 3 - m 2 ) + c 2 (m 1 - m 3 ) + c 3 (m 2 - mf)} 2 . 
(m 3 - m 2 )(m 1 - m 3 )(m 2 - mf 
We may further observe, that if the equations to the sides of ABC 
be given in the form ax + by c — 0, the corresponding expressions 
for twice the area of the triangle are respectively 
(^3^2 ~ ^2^3)^ _j_ (^1^3 ~ ^3^l) 2 _j_ (^2^1 
bffafc} — d 2 b 3 ) b 3 b 1 (a 1 b 3 — afi) b 1 b 2 ( y a 2 b 1 — uq& 2 ) 
and 
( c 3 a 2 - c 2 a 3 ) 2 (c 1 c * 3 - c 3 a A ) 2 (c 2 oq - c 1 cx 2 ) 2 
d 2 Cl 3 (cif) 2 — <X 2 & 3 ) ClyCl-^CL-fs ~ ^sPi) d^€l 2 (CL 2 b^ — df 2 ) 
