Hogg. — On certain Tripolar Relations. 321 



Nine-point circle, p = 4R 



a cos (B - C) X ~+ b cos (C - A) Y + c cos (A - B) Z 



= abc (1 + 2 cos A cos B cos C). 



B, 2 

 Orthocentric circle, p 2 = — (1 — 8 cos A cos B cos C) 



a cos (B - C) X + b cos (G - A) Y + c cos (A - B) Z = abc. 



9. Circles of Appolonius 



b 2 Y - c 2 Z = o, c 2 Z - a 2 X = o, a 2 X - 6 2 Y = o. 



10. Circles with centres at Brocard points 



fi(x> ->-)i fi' (-,-,?), radius = p 

 \o c aj \c a bj 



c 2 a 2 X + a 2 6 2 Y + 6VZ = a 2 6V + 2 (^V 2 ) P 2 

 6 2 a 2 X + c 2 6 2 Y + aVZ - a 2 6V + 2 (6V) P 2 . 



11. Circle having fiO' as diameter 



a 2 (b 2 + c 2 ) X + 6 2 (c 2 + a 2 ) Y + c 2 (a 2 + 6 2 ) Z = a 2 b 2 c 2 (1 + 2 cos 2o>), 

 where w is the Brocard angle of the triangle. 



12. Brocard circle 



abc 2 (X) + 2 (a 8 cos A X) = \abc 2 (a 2 ). 



13. Circle on IJ X as diameter 



- a 2 X + (b + c) (6Y + cZ) = a 2 bc. 



14. Circle on I 2 I 3 as diameter 



a 2 X - (b - c) (6Y - cZ) - a 2 6c. 



15. Circle having the bisector of the angle A as diameter 



{b + c) X + bY + cZ = 6c (6 + c). 



16. Circle having side of pedal triangle as diameter 



a 2 sec A cos (B - C) X + 6 2 Y + c 2 Z = 8 a 2 . 



17. Circle on BC containing angle A. 



a cos A. X — b cos (c + A.) Y — c cos (B + A) Z = a&c cos (A — A). 



18. Polar circles of the triangles BOC, COA, AOB, where is the 

 orthocentre of the triangle ABC 



X = 26ocos A. Y = 2ca cos B, Z = lab cos C. 



The tripolar equation of the Nine-point circle may be obtained in the 

 following manner. 



Let X', Y', Z' be the squares of the distances of any point in the plane 

 of the triangle ABC from the mid-points of the sides of that triangle. 



The Nine-point circle being the circumcircle of the medial triangle, its 

 equation is 



- cos A X' + - cos B Y' + - cos C Z' = — 

 2 2 2 8 



Also Y + Z = 2X' + -, Z + X = 2Y' + -, and X + Y = 2Z' +_ 



2 2 2 



Hence 2 {a cos A (Y + Z) } = h {abc + 2 (a 3 cos A)} 



from which we obtain 



2 | a cos (B - C) X } = abc (1 + 2 cos A cos B cos C) 

 11— Trans. 



