(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. 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 sqrt(p) + b sqrt(q) + c sqrt(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 [rho] be the radius and lp + mq + nr = 0 the centre, we have [rho] = (lp1 + mq1 + nr1)/(l + m + n), in which p1, q1, r1 is a line distant [rho] from the point lp + mq + nr = 0. Making this equation homogeneous by the relation [Sigma]a²(p - q) (p - r) = 4[Delta]² (see GEOMETRY: _Analytical_), which is generally written { ap, bq, cr}² = 4[Delta]², we obtain { ap, bq, cr}²[rho]² = 4[Delta]²{ (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 = ([lambda]p + [mu]q + [nu]r)², obviously represents a circle, the centre being [lambda]p + [mu]q + [nu]r = 0, and radius 2[Delta]/([lambda] + [mu] + [nu]). If we make [lambda] = [mu] = [nu] = 0, [rho] is infinite, and we obtain { ap, bq, cr}² = 0 as the equation to the circular points.
_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.