88 



22. If from K',i perpendiculars p an q are drawn to CA, AB respectively, 

 then ^ 



p q _ 2 A 



b ~~ c a^-fb^ 



23. AKiKK'a — bM-o2:a2 



24. BK'a:K'aC=:c2:b2 

 CK^b-.K'b A = a2:c2 

 AK^c:K^cB = b2:a2 



BK^„ = --^ etc 



o- -j-c- 



25. The tangent to the circumcircle at A, and the symmedian AK are har- 

 monic conjugates with respect to AB and AC. - 



26. The angles AMK, BMK, CMK are equal respectively to the angles 

 (BC, BiCi) (AC, AiCi), (AB, AiBi), that is the respective angles between the 

 sides of Brocard's first triangle and the corresponding sides of the fundamental 

 triangle. 



27. The sides of the J\KaKbKc are proportional to the medians of the 

 AA.BC, and the angles of the AKaKbKc are equal to the angles which the 

 medians make with each other. 



28. The Bum of the squares of the sides of KaKbKc is less than the sum of 

 the squares of the sides of any other triangle inscribed in ABC. 



29. The ratio of the area of ABC to that of its co-symmedian triangle 

 A''B'C^ (See No. 10) is (— a2-f2b2+2c2) (2a2— b24-2c=') (2a2+2b2— c2):27a2bV. 



Note On McGinnis's Universal Solution. 



By Robert J. Aley. 



The full title of the book is, "The Universal Solution for numerical 

 and literal equations by which the roots of equations of all degrees can 

 be expressed in terms of their coefficients, by M. A. McGinnis, Kansas 

 City, Missouri, the Mathematical Book Company, 1900." 



In his preface the author announces that the book appears at "the 

 request of many able mathematicians, teachers and scholars throughout 

 the United States.'" He also modestly states that the imaginary is for 

 the first time put upon a true basis, that bi-quadratics are more thoroughly 



