MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 67 



$(a, 6, Cj rj, rj, j-g). (47) 



Lemma. If a, p are any two segments terminating at P and 8 is their 

 resultant, the lines of action cut any circle through P in three points 

 A, B, C respectively distant r^, i^, 1\ from p, then 



S 7-3 = a 1\ -I- p r,. 

 This relation is undisturbed, if A, B, C are fixed points and P is moved 

 along the circle to coincide with A or B, a, p, S being constant. 



Consider the problem of § 3 as solved. Construct circles AP B, B P C, 

 C P A. Produce AP, BP, GP to meet these circumferences respectively 

 again in A^, B^, C^. Construct the triangle A B C^, then by the lemma 



««'i + /8r„— y.FCi = 0. (48) 



Move the point P along the arc first to coincidence with A, then with B, 

 whence 



AC, = ^c, BC,= — c* 

 Y 7 



The triangle A B C-^is. completely known. The radial coordinates 



^c,^c, CO,, 

 7 7 



of Ci with respect to A B C satisfy the radial identity 



/3c 



0. 



* (a, 6, c, -^^^-^^^, OCJ 

 7 7 

 Hence C C^ is determined by a quadratic equation. Also by (32), 



a . A A^ ^ p . B B, = y . C C^ = s, 

 therefore s is known. 



Thence directly the power p^ of C with respect to the circle A B C^ie 



(49) 



*We may observe that a, ^, 7, are the coordinates (with respect to ABd) 

 of an excenter of ABC, since they are proportional to the sides of that triangle, 

 and (48) is the ptolemaic equation (in radial coordinates with respect to ABCi) 

 of the arc APB. 



