ng 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] squared + v[beta] squared + w[gamma] squared
+ 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 = vc squared + wb squared - 2u'bc = wa squared + uc squared - 2v'ca = ub squared + va squared - 2w'ab.

Areal co-ordinates.

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 a squaredyz + b squaredzx + c squaredxy = 0, with centre sin 2A, sin 2B, sin 2C; the
inscribed circle is sqrt(x cot 1/2A) + sqrt(y cot 1/2B) + sqrt(z cot 1/2C) =
0, with centre sin A, sin B, sin C; the escribed circle opposite the
angle A is sqrt(-x cot 1/2A) + sqrt(y tan 1/2B) + sqrt(z tan 1/2C)=0, with
centre - sin A, sin B, sin C; and the self-conjugate circle is x squared cot
A + y squared cot B + z squared 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 a squaredyz + b squaredzx +
c squaredxy + (lx + my + nz)(x + y + z) = 0. Comparing this equation with ux squared
+ vy squared + wz squared + 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')/a squared = (w + u - 2v')/b squared = (u + v - 2w')/c squared.

Tangential co-ordinates.

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 1/2(a +
b + c); an alternative form is qr cot 1/2A + rp cot 1/2B + pq cot 1/2C = 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 1/2A + rp tan 1/2B + pq tan 1/2C = 0, with ce