Miscellaneous Exercises on the Circle. 149

The co-ordinates of the radical centres of the three escribed circles are
ri cos | (B — C) 1 2 sin f A, &c. Substitute these in the equation of the ex-
circle, which touches a externally, viz. —

a2 cos4f ^ + /32sin4J£ + 72 sin* } C' - 2/3y sin2 f .B sin2f (7

+ 2y a sin2 J (7 cos2 %A + 2aj3 cos2 f ^ sin2 J .5,
and divide the result hy the modulus of the circle ; that is, by

4 cos2 j- .4 sin2 J 2? sin2 J £

The quotient is the square of the radius of the orthogonal circle. In
reducing, we substitute for r the value a sin J 2? sin JCycosJ-4. Thus
we get —

« i

COS 5 H- COSj5 COS 0+

61. If A', £', Cr be the feet of the altitudes of the triangle A£C, prove
that the joins of the incentre and circumcentre of the triangles AB'C',
SO' A', CA'B', respectively, are concurrent, and that the common point is
at the contact of the incirole and " Nine-points Circle."

62. A similar theorem is true for the joins of the excentres and circum-
centres.

63. The diameters of the circles cutting the inscribed circle and two
escribed circles orthogonally are

-A-7 (1 -1- cos A cos £ - cos B cos C + cos 0 cos A)$, &c. (361)

64. Prove by the modulus of the equation of the " Nine-points Circle "
that it touches the inscribed and escribed circles.

65. Prove that the determinant

y+f",
y+f",

: 0 (362)

is the circle orthogonal to the three circles a?2 + y2 + 2^'^ + 2fy + cr = 0,
&c.
66. There exists a relation of the form 2wP= constant, where m\t mz, &c.,
are certain constants whose sum is zero, between the powers PI, JPz, &c.,
of any arbitrary point If, and four fixed circles whose centres are A\t A»,
&C. (LUCAS.)