iO Transactions. 



5. Let D, E, F be respectively the middle points of BC, CA, and AB : 

 then the equations of EF, FD, and DE are 



— aa + hjB -\- Cy = ... ... ... ... (i) 



Oa — b/3 + Cy = ... ... ... ... (ii) 



aa + b/^ — cy = ... ... ... .. (iii) 



The line ^ — y ■= o will meet the first of these lines in the point 

 [h + c, a, a) : using this point as centrum we have the conic 



a(a'- + ^y) - (& + c)a(/3 + y) =0 E 



which meets the circle ABC along the line 



aa- {h + 2c) yS - (c + 26) y = o, 

 or 



aa + &/3 + cy - 2 (/j + c) (^ + y) = 0. 



This line meets the internal bisector of angle A at the point [3(6 + c), a, a\ , 

 which is the middle point of the line joining A to the point in which the 

 internal bisector of A meets EF. Hence the conic is a hyperbola, 

 whose centre and asymptotes are found in the manner previously 

 employed. 



The line ^ — y = o meets the lines (ii) and (iii) respectively in the 

 points [{b — c), a, a] and [—{b — c), a, a]. With these points as centra 

 we obtain the conies 



a{a'+ f3y)~ {b-c)a{ft + y) = F 



a{a' + (Sy) + ib-c)a(J3+y) =0 G. 



The former of these conies meets the circle along the line 



aa — b(3 — (26 — c) y = 0, 



which is parallel to /? + y = o and passes through the middle point of 

 AB. The latter conic meets the circle ABC along the line 



aa + {b - 2c) fS - Cy = 0, 



a line parallel to /S + y = o and passing through the middle point of AC. 

 Hence each of the conies F and G is a hyperbola. 



6. The line /? + y = o will meet the lines (i), (ii), and (iii) of the 

 preceding section respectively in the points 



[{b - c), a, -a], [{b + c), a, -a], [{b + c), -a, a]. 



Using these points as centra we have the conies 



aid" - I3y) - (b - C)a{(3 -y) = H 



a(a--y8y) - (6 + c)a(/S -y) =0 J 



a{a:'- fSy)-t{b + c)a{(3-y)^0 K. 



Their chords of intersection with the circle ABC are respectively 



aa + (2c - 6) /3 + (26 - c) y = 



aa - b/3 + {2b + c) y = 



aa -j- (6 + 2c) p — Cy = 0. 



These lines are all parallel to the internal bisector of the angle A : the 

 first of them passes through the point [3(6 — c),a, —a], w^hich is the 

 middle point of the line joining A to the point in which the external 

 bisector of A meets EF. The second and third lines pass respectively 

 through the middle points of AB and AC. 



