Circumcircle

A Triangle's circumscribed circle. Its center $O$ is called the Circumcenter, and its Radius $R$ the Circumradius. The circumcircle can be specified using Trilinear Coordinates as

$$\beta\gamma a+\gamma\alpha b+\alpha\beta c=0.$$ (1)

The Steiner Point $S$ and Tarry Point $T$ lie on the circumcircle.

A Geometric Construction for the circumcircle is given by Pedoe (1995, pp. xii-xiii). The equation for the circumcircle of the Triangle with Vertices $(x_i, y_i)$ for $i=1$, 2, 3 is

$$\left\vert\matrix{ x^2+y^2 & x & y & 1\cr {x_1}^2+{y_1}^2 & ... ...& y_2 & 1\cr {x_3}^2+{y_3}^2 & x_3 & y_3 & 1\cr}\right\vert=0.$$ (2)

Expanding the Determinant,

$$a(x^2+y^2)+2dx+2fy+g=0,$$ (3)

where

$$a = \left\vert\begin{array}{ccc}x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1\end{array}\right\vert$$ (4)

$$d = -{\textstyle{1\over 2}}\left\vert\begin{array}{ccc}{x_1}^2+{y_1}^... ...\ {x_2}^2+{y_2}^2 & y_2 & 1\\ {x_3}^2+{y_3}^2 & y_3 & 1\end{array}\right\vert$$ (5)

$$f = {\textstyle{1\over 2}}\left\vert\begin{array}{ccc}{x_1}^2+{y_1}^2... ...\ {x_2}^2+{y_2}^2 & x_2 & 1\\ {x_3}^2+{y_3}^2 & x_3 & 1\end{array}\right\vert$$ (6)

$$g = -\left\vert\begin{array}{ccc}{x_1}^2+{y_1}^2 & x_1 & y_1\\ {x_2}^2+{y_2}^2 & x_2 & y_2\\ {x_3}^2+{y_3}^2 & x_3 & y_3\end{array}\right\vert.$$ (7)

Completing the Square gives

$$a\left({x+{d\over a}}\right)^2+a\left({y+{f\over a}}\right)^2-{d^2\over a}-{f^2\over a}+g=0$$ (8)

which is a Circle of the form

$$(x-x_0)^2+(y-y_0)^2=r^2,$$ (9)

with Circumcenter

$$x_0 = -{d\over a}$$ (10)

$$y_0 = -{f\over a}$$ (11)

and Circumradius

$$r=\sqrt{{f^2+d^2\over a^2}-{g\over a}}.$$ (12)

See also Circle, Circumcenter, Circumradius, Excircle, Incircle, Parry Point, Purser's Theorem, Steiner Points, Tarry Point

References

Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.