318 Transactions. 



The following particular cases of (iv) are of interest. For the ortho- 

 centre, in-centre, centroid, and symmedian point respectively 



sin A cos A + siu B cos p + sin C cos v = o 



A ^ B C 



COS -- COS A + COS — COS p + COS .— COS v — 



A A A 



m r COS A + IU. 2 COS p + ^h COS V = o 



am,! cos A + bin., cos p + cm ?, cos v — o, 

 where m 1( m z , and m 3 are the medians of the triangle ABC. 



If be either of the two points whose pedal triangles are equiangular, 

 then, since for these points ap 1 = bp. 2 = cp A , 



a Q COS A + (S COS p + y COS v = 0. 



If be the focus of a conic inscribed in the triangle ABC, then the 

 equation of the conic is 



a 



\/X a a + b VT of 3 o f3 + c v Z o7o7 = o. 



Comparing this with the equation of the maximum inscribed ellipse, 

 v aa + Vb(3 4 v ' cy = o, we have aX a = bY /3 ■-= cZ y , whence 



cos A cos p, cos v 

 AO~ + ~B0~ + "CO" = °' 



For the Brocard ellipse this gives 



COS A COS fi. cos v 



BC . AO ' CA . BO ' AB . CO 



Let U 1 be the result of substituting in U the co-ordinates (X 1 , Y 1 , Z 1 ) 

 for (X, Y, Z). Suppose a circle of radius p 1 concentric with U to pass 

 through the point (X^Z 1 ), then 



U 1 = aa Q X l + 6&Y 1 + cy.Z 1 - 2ES - 2a P 2 



= aa Q X l + 6/3 Q Y' + cy^ 1 - 2BS - 2 a p 1 2 



.-. U 1 = 2 a (p 1 a - p~) = 2 a t 1 2 , 



where t 1 is the length of the tangent from (X^Z 1 ) to the circle U = O. 



Let t lt t.,, and t s be the lengths of the tangents to the circle U from 

 A, B, C respectively : then 



2 a t? = b/3 c 2 + cyjb 1 - 2 a (B 2 - cV- + p 2 ) ; 



2 a (tf + B 2 - d. 2 + p 2 ) = 6y8 c 2 + cy b\ 



We have X = t£ + p' 2 , whence, if OH subtend at A, B, and C the 

 angles 0, <j>, </, respectively, 



abc pj cos 6 = bfi c- + cy 6' 2 



abc p<, cos <£ = aa Q c 2 + cy a 2 



abc p :i cos i/' = aajf -f- 6/? a. 2 



2 a - aa + ty3 + Cy. 



