This obviously represents a conic intersecting the circle aβγ + bγα + cαβ = 0 in points on the common chords lα + mβ + nγ = 0, aα + bβ + cγ = 0. The line lα + mβ + nγ is the radical axis, and since aα + bβ + cγ = 0 is the line infinity, it is obvious that equation (1) represents a conic passing through the circular points, i.e. a circle. If we compare (1) with the general equation of the second degree uα² + vβ² + wγ² + 2u′βγ + 2v′γα + 2w′αβ = 0, it is readily seen that for this equation to represent a circle we must have
-kabc = vc² + wb² - 2u′bc = wa² + uc² - 2v′ca = ub² + va² - 2w′ab.
The corresponding equations in areal co-ordinates are readily derived by substituting x/a, y/b, z/c for α, β, γ respectively in the trilinear equations. The circumcircle is thus seen Areal co-ordinates. to be a²yz + b²zx + c²xy = 0, with centre sin 2A, sin 2B, sin 2C; the inscribed circle is √(x cot ½A) + √(y cot ½B) + √(z cot ½C) = 0, with centre sin A, sin B, sin C; the escribed circle opposite the angle A is √(-x cot ½A) + √(y tan ½B) + √(z tan ½C)=0, with centre - sin A, sin B, sin C; and the self-conjugate circle is x² cot A + y² cot B + z² cot C = 0, with centre tan A, tan B, tan C. Since in areal co-ordinates the line infinity is represented by the equation x + y + z = 0 it is seen that every circle is of the form a²yz + b²zx + c²xy + (lx + my + nz)(x + y + z) = 0. Comparing this equation with ux² + vy² + wz² + 2u′yz + 2v′zx + 2w′xy = 0, we obtain as the condition for the general equation of the second degree to represent a circle:—
(v + w - 2u′)/a² = (w + u - 2v′)/b² = (u + v - 2w′)/c².
In tangential (p, q, r) co-ordinates the inscribed circle has for its equation (s - a)qr + (s - b)rp + (s - c)pq = 0, s being equal to ½(a + b + c); an alternative form is qr cot ½A + rp cot ½B + pq cot ½C = 0; the centre is ap + bq + cr = 0, or p sin A + q sin B + r sin C = 0. Tangential co-ordinates. The escribed circle opposite the angle A is -sqr + (s - c)rp + (s - b)pq = 0 or -qr cot ½A + rp tan ½B + pq tan ½C = 0, with centre -ap + bq + cr = 0. The circumcircle is a √(p) + b √(q) + c √(r) = 0, the centre being p sin 2A + q sin 2B + r sin 2C = 0. The general equation to a circle in this system of co-ordinates is deduced as follows: If ρ be the radius and lp + mq + nr = 0 the centre, we have ρ = (lp1 - mq1 + nr1/(l + m + n), in which p1, q1, r1 is a line distant ρ from the point lp + mq + nr = 0. Making this equation homogeneous by the relation Σa²(p - q) (p - r) = 4Δ² (see [Geometry]: Analytical), which is generally written {ap, bq, cr}² = 4Δ², we obtain {ap, bq, cr}²ρ² = 4Δ² {(lp + mq + nr)/(l + m + n)}², the accents being dropped, and p, q, r regarded as current co-ordinates. This equation, which may be more conveniently written {ap, bq, cr}² = (λp + μq + νr)², obviously represents a circle, the centre being λp + μq + νr = 0, and radius 2Δ/(λ + μ + ν). If we make λ = μ = ν = 0, ρ is infinite, and we obtain {ap, bq, cr}² = 0 as the equation to the circular points.
Systems of Circles.
Centres and Circle of Similitude.—The “centres of similitude” of two circles may be defined as the intersections of the common tangents to the two circles, the direct common tangents giving rise to the “external centre,” the transverse tangents to the “internal centre.” It may be readily shown that the external and internal centres are the points where the line joining the centres of the two circles is divided externally and internally in the ratio of their radii.
The circle on the line joining the internal and external centres of similitude as diameter is named the “circle of similitude.” It may be shown to be the locus of the vertex of the triangle which has for its base the distance between the centres of the circles and the ratio of the remaining sides equal to the ratio of the radii of the two circles.
With a system of three circles it is readily seen that there are six centres of similitude, viz. two for each pair of circles, and it may be shown that these lie three by three on four lines, named the “axes of similitude.” The collinear centres are the three sets of one external and two internal centres, and the three external centres.
Coaxal Circles.—A system of circles is coaxal when the locus of points from which tangents to the circles are equal is a straight line. Consider the case of two circles, and in the first place suppose them to intersect in two real points A and B. Then by Euclid iii. 36 it is seen that the line joining the points A and B is the locus of the intersection of equal tangents, for if P be any point on AB and PC and PD the tangents to the circles, then PA·PB = PC² = PD², and therefore PC = PD. Furthermore it is seen that AB is perpendicular to the line joining the centres, and divides it in the ratio of the squares of the radii. The line AB is termed the “radical axis.” A system coaxal with the two given circles is readily constructed by describing circles through the common points on the radical axis and any third point; the minimum circle of the system is obviously that which has the common chord of intersection for diameter, the maximum is the radical axis—considered as a circle of infinite radius. In the case of two non-intersecting circles it may be shown that the radical axis has the same metrical relations to the line of centres.