The equation to the circumcircle assumes the simple form a[beta][gamma] + b[gamma][alpha] + c[alpha][beta] = 0, the centre being cos A, cos B, cos C. The inscribed circle is cos ½A sqrt([alpha]) cos ½B sqrt([beta]) + cos ½C sqrt([gamma]) = 0, with centre [alpha] = [beta] = [gamma]; while the escribed circle opposite the angle A is cos ½A sqrt(-[alpha]) + sin ½B sqrt([beta]) + sin ½C sqrt([gamma]) = 0, with centre -[alpha] = [beta] = [gamma]. The self-conjugate circle is [alpha]² sin 2A + [beta]² sin 2B + [gamma]² sin 2C = 0, or the equivalent form a cos A [alpha]² + b cos B [beta]² + c cos C [gamma]² = 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
a[beta][gamma] + b[gamma][alpha] + C[alpha][beta] + (l[alpha] + m[beta] + n[gamma]) (a[alpha] + b[beta] + c[gamma]) = 0 (1).
This obviously represents a conic intersecting the circle a[beta][gamma] + b[gamma][alpha] + c[alpha][beta] = 0 in points on the common chords l[alpha] + m[beta] + n[gamma] = 0, a[alpha] + b[beta] + c[gamma] = 0. The line l[alpha] + m[beta] + n[gamma] is the radical axis, and since a[alpha] + b[beta] + c[gamma] = 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 u[alpha]² + v[beta]² + w[gamma]² + 2u'[beta][gamma] + 2v'[gamma][alpha] + 2w'[alpha][beta] = 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 [alpha], [beta], [gamma] respectively in the trilinear equations. The circumcircle is thus seen to be a²yz + b²zx + c²xy = 0, with centre sin 2A, sin 2B, sin 2C; the inscribed circle is sqrt(x cot ½A) + sqrt(y cot ½B) + sqrt(z cot ½C) = 0, with centre sin A, sin B, sin C; the escribed circle opposite the angle A is sqrt(-x cot ½A) + sqrt(y tan ½B) + sqrt(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:--