320 Transactions. 



If ia the above equation [K cos A, R cos B, R cos C] be substituted 

 for (a /3 o y ) and p be made equal to R, we obtain for the equation of the 

 circle ABC 



a cos A X + b cosB Y + c cos B Z — abc = o (3) 



If the square of (3) be subtracted from the fundamental relation con- 

 necting four coplanar points, viz., 



2(a 2 X 2 ) - 22 {be cos A YZ) - <2abc 2 (a cos A X) + a 2 b' 2 c 2 = o, (4) 



we have 2 (a 4 X a ) - 22 (6VYZ) = o, 



whence 



(«X* + 6Y* + cZi) (- aX* + 6Y* + cZ*) (aX* - bYi + cZi) 

 {aXi + 6Y* - cZ*) = o, 

 a known relation connecting the distances of any point on a circle from 

 the vertices of an inscribed triangle. 



The following list contains the equations of some of the more 

 important circles connected with the triangle. 



1. Centre at circumcentre, radius = p 



a cos A X + b cosB Y + c oosC Z = 4R sin A sinB sin C (R 2 + p 2 ). 



2. Centre at orthocentre, radius = p 



tan A X + tan B Y + tan C Z = p 2 tan A tan B tan C + 4 a . 



In-circle of pedal triangle, p = 2R cos A cos B cos C 



tan A X + tanB Y + tanC Z = *R 2 sin 2 A sin2B sin2C + 4 a. 



Polar circle, p 2 — — 4R 2 cos A cos B cos C 

 tan A X + tan B Y + tan C Z = 2 a . 



3. Centre at centroid, radius = p 



X + Y + Z = ±2 (ft 2 ) + 3p 2 . 



4. Centre at symmedian point, radius = p 



a>X + b*Y + c 2 Z = ^tlL + 2 (a 2 ) p\ 



2 (a 2 ) 



5. Centre at in-centre, radius = p 



aX + 6Y + cZ = (a + 6 + c) (2Rr + p 2 ). 

 In-circle, p = r 



aX + bY + cZ = (a + b + c) (2Rr + r 2 ). 



6. Centre at ex-centre (— 111), radius = p 



aX - bY - cZ = (b + c - a) (2Rr, - p a ). 



Ex-circle, p = r a 



a X - />Y - cZ = (b + c - a) (2Br, - ft*). 



7. Circle concentric with circle IiI 2 I 3 , radius = p 



a (2R - r t ) X + 6 (2R - r a ) Y + c (2R - r 3 ) Z = «&cr + 2 Ap 2 . 



Circle I, I, I 8> p = 2R 

 a (2R - r x ) X + 6 (2R - r 2 ) Y + c (2R - r 8 ) Z = a&c (2R + r). 



8. Centre at Nme-point centre, radius = p 



R [a cos (B - C) X + b cos (C - A) Y + c cos (A - B) Z] 

 = R 2 A (3 + 8 cos A cos B cos C) + 4Ap 2 . 



