Analytical Geometry of the Circle.
In the article [Geometry]: Analytical, it is shown that the general equation to a circle in rectangular Cartesian co-ordinates is x2 + y2 + 2gx + 2fy + c = 0, i.e. in the general equation of the second degree the co-efficients of x2 and y2 are Cartesian co-ordinates. equal, and of xy zero. The co-ordinates of its centre are -g/c, -f/c; and its radius is (g2 + f2 - c)½. The equations to the chord, tangent and normal are readily derived by the ordinary methods.
Consider the two circles:—
x2 + y2 + 2gx + 2fy + c = 0, x2 + y2 + 2g′x + 2f′y + c’ = 0.
Obviously these equations show that the curves intersect in four points, two of which lie on the intersection of the line, 2(g - g′)x + 2(f - f′)y + c - c′ = 0, the radical axis, with the circles, and the other two where the lines x² + y² = (x + iy) (x - iy) = 0 (where i = √-1) intersect the circles. The first pair of intersections may be either real or imaginary; we proceed to discuss the second pair.
The equation x² + y² = 0 denotes a pair of perpendicular imaginary lines; it follows, therefore, that circles always intersect in two imaginary points at infinity along these lines, and since the terms x² + y² occur in the equation of every circle, it is seen that all circles pass through two fixed points at infinity. The introduction of these lines and points constitutes a striking achievement in geometry, and from their association with circles they have been named the “circular lines” and “circular points.” Other names for the circular lines are “circulars” or “isotropic lines.” Since the equation to a circle of zero radius is x² + y² = 0, i.e. identical with the circular lines, it follows that this circle consists of a real point and the two imaginary lines; conversely, the circular lines are both a pair of lines and a circle. A further deduction from the principle of continuity follows by considering the intersections of concentric circles. The equations to such circles may be expressed in the form x² + y² = α², x² + y² = β². These equations show that the circles touch where they intersect the lines x² + y² = 0, i.e. concentric circles have double contact at the circular points, the chord of contact being the line at infinity.
In various systems of triangular co-ordinates the equations to circles specially related to the triangle of reference assume comparatively simple forms; consequently they provide elegant algebraical demonstrations of properties concerning a triangle and the circles intimately associated with its geometry. In this article the equations to the more important circles—the circumscribed, inscribed, escribed, self-conjugate—will be given; reference should be made to the article [Triangle] for the consideration of other circles (nine-point, Brocard, Lemoine, &c.); while in the article [Geometry]: Analytical, the principles of the different systems are discussed.
The equation to the circumcircle assumes the simple form aβγ + bγα + cαβ = 0, the centre being cos A, cos B, cos C. The inscribed circle is cos ½A √(α) cos ½B √(β) + cos ½C √(γ) = 0, with centre α = β = γ; while the escribed circle opposite the angle A Trilinear co-ordinates. is cos ½A √(-α) + sin ½B √(β) + sin ½C √(γ) = 0, with centre -α = β = γ. The self-conjugate circle is α² sin 2A + β² sin 2B + γ² sin 2C = 0, or the equivalent form a cosA α² + b cos B β² + c cos C γ² = 0, the centre being sec A, sec B, sec C.
The general equation to the circle in trilinear co-ordinates is readily deduced from the fact that the circle is the only curve which intersects the line infinity in the circular points. Consider the equation
aβγ + bγα + Cαβ + (lα + mβ + nγ) (aα + bβ + cγ) = 0 (1).