158 BOOK OP MATlir.MATHAL PROBLEMS. 



3. The conic fa* + mfl* + ny* = 

 will represent a circle if 



/ tan A = m tan B = n tan C. 



The necessary and sufficient conditions that the equation 



la* + m(P + wy* + I'fiy + mya + n'afl = 

 may represent a circle are 



m + n-t _n + l-mf _l + m-n' 



The lengths of the tangents from -4, B, C to a circle are 

 p, q t r j prove that the equation of the circle is 



a'/?y + b*ya + c'a/J = (a + )8 + y) (p 3 a + q*j3 + r'y). 



756. P, P,, P,, P 8 are the points of contact of the Nine 

 Points' Circle with the inscribed and escribed circles ; prove that 

 (1) the equations of the tangents at these points are 



) 



o c ca ab 



and the three equations obtained from this by changing the sign 

 of a, 6, or c; (2) PP,, P 8 P 8 meet EG in the same points as the 

 straight lines bisecting the internal and external angles at A\ 



(3) PP,, P 8 P a intersect in the point 



- P y 



6>_ c c *- a a ~a a -b*' 



(4) the tangents at P, P l9 P a , P a all touch the ellipse which 

 touches the sides of the triangle at their middle points. 



757. The straight line la + m/3 + ny = Q meets the sides of the 

 triangle ABC in A', ', C' ; prove that the circles on A A', BE', 

 CC' have the common radical axis 



a cot A (!- l } +p cot E (- - ]} + y cot C (\ - -V : 

 \m n) \n // ' \J m) 



and the circles will touch each other if 



'(a 4 + -26V- ) 



= 8lmn{la 4 + - (m + n) 6V - }. 



