87 



6. The line KMa (Ma is the mid point of BC) passes through the mid point 

 (>f the altitude AHa. 



7. The sides of the K-pedal triangle KaKbKc are perpendicular to the 

 medians of ABC, respectively. 



8. The sides of the G-pedal triangle GaGbGc are perpendicular to the sym- 

 medians AK, BK, CK, respectively, 



9. a.GA.KA + b.GB. KB + C.GC. KC^^a.b.c. 



10. If the symmedian lines AK, BK, CK meet the circumcircle of ABC in 

 A^, B', C, then the triangles ABC and A'B'C are co-symmedian, that is they 

 have the same symmedian point K. 



11. K and M (M is the circumcentre of ABC) are opposite ends of a diam- 

 eter of Brocard's Circle. 



12. Parallels to the sides of ABC through K, determine six points on the 

 sides which lie on the Lemoine Circle. 



13. If points A^ B^ C" be taken on KA, KB, KC so that KA' : KB' : KC = 

 KA : KB : KC ::= constant, then antiparallels to the sides through A', B', C, re- 

 spectively, determine six points on the sides of the triangle which lie on a Tucker 

 Circle. 



14. If Ai Bi Ci is Brocard's first triangle, then 



Ai K is parallel to BC. 

 Bi K is parallel to CA. 

 Ci K is parallel to AB. 



15. AK, BK, CK produced meet Brocard's circle again in A'^, B'^, C re- 

 spectively, and these points form Brocard's second triangle k." W Q,'\ 



16. If KA, KB, KC, meet the sides of ABC in Xi, X2, Yi, Y2 and Zi, Z2 

 respectively, then the sides of the triangle Zi Xi Yi are parallel to A i2, B ^2, C i2 

 respectively, and the sides of Y2 Z2 X2 are parallel to A i2', B i2', C il^ respectively, 

 where O and Q.' are the Brocard points of ABC. S2 and K are the Brocard points 

 of ZiXiYi and SI' and K are the Brocard points of Y2Z2X2. 



17. The point of concurrency' D of AAi, BBi, CCi is the point isotomic 

 conjugate to K. 



18. The line MK is perpendicular to and bisects the line 1212'. 



19. The Simson line of Tarry's point is perpendicular to MK. 



20. (Jot <KBC+ cot <KCA + cot <KAB = 3 cot (.^ where u is the Bocard 

 angle. 



21. If the symmedian AK cut BC in K'a and the line MMa in Q then 

 (AK'a, KQ) is a harmonic range. 



