MAXIMUM AND MINIMUM VALUES OF A LINEAR FUNCTION, ETC. 01 



G. The relations between the angles involved in the previous discus- 

 sion are as follows : Let ©, *, * be the absolute angles subtended at F by 

 the sides a, b, c respectively. Then when Q and P are inside ABC, 



{r,r,)=APB = ^=:'K— C, 

 {i\ r^)= B P C = ® = w —A' , 

 {rlr^)=CPA = ^ = n — B'. 



AVhen P is outside and in angle C 



( rj r, ) = 4 P 5 = — * = C— w, 

 (r^r^)=BPC= © = !', 

 {r,r^)=CPA= # = B'. 



"V\Tien P is in vertical angle C 



{r-,r„)=APB= * = ^_C", 

 (r^r^)=BPC = ~®= —A', 

 (r,r,)=CPA= — ^= —5'. 



Similar results hold when P is outside and in angles B, A or their verticals. 

 Hence in all cases 



cot APB= z^ cot C", cot 5 P C = q= cot A', cot C P .4 = =j= cot B', (15) 



the upper signs being taken when P is inside ABC and the lower when 

 outside. 



7. Directly from the figure 



ri^-|- r,^ — 2 r^ Tr, cos A P B = c^, "1 



r/+r,^—2i\r^cosBPC = a\ ^ (16) 



^3"+ r^-— 2 r, i\ cosCPA = l^. J 



There are also the known relations 



?•-,- = c-y -)- 6-2 — Lxyc^, 1 



r,^ = c- a- -)- a- a — 'S,xyc-, r (17) 



rg- = 6- .T -\- a- y — 'S, x y c-, ^ 



expressing the squares of the distances of a point from the corners of the 



triangle of reference A B C in terms of the areal coordinates of the point. 



Equations (16) are the equations in radial coordinates of the circles 



BPC, C P B, BPA. Eliminating the radial coordinates between (10), 



