t 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
x squared + y squared = [alpha] squared, x squared + y squared = [beta] squared. These equations show that the
circles touch where they intersect the lines x squared + y squared = 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.

Trilinear co-ordinates.

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 1/2A sqrt([alpha])
cos 1/2B sqrt([beta]) + cos 1/2C sqrt([gamma]) = 0, with centre [alpha] =
[beta] = [gamma]; while the escribed circle opposite the angle A is
cos 1/2A sqrt(-[alpha]) + sin 1/2B sqrt([beta]) + sin 1/2C sqrt([gamma]) =
0, with centre -[alpha] = [beta] = [gamma]. The self-conjugate circle
is [alpha] squared sin 2A + [beta] squared sin 2B + [gamma] squared sin 2C = 0, or the
equivalent form a cos A [alpha] squared + b cos B [beta] squared + c cos C [gamma] squared =
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 intersecti