info prev up next book cdrom email home

Circumcircle

\begin{figure}\begin{center}\BoxedEPSF{Circumcircle.epsf scaled 800}\end{center}\end{figure}

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

\begin{displaymath}
\beta\gamma a+\gamma\alpha b+\alpha\beta c=0.
\end{displaymath} (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

\begin{displaymath}
\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.
\end{displaymath} (2)

Expanding the Determinant,
\begin{displaymath}
a(x^2+y^2)+2dx+2fy+g=0,
\end{displaymath} (3)

where
$\displaystyle a$ $\textstyle =$ $\displaystyle \left\vert\begin{array}{ccc}x_1 & y_1 & 1\\  x_2 & y_2 & 1\\  x_3 & y_3 & 1\end{array}\right\vert$ (4)
$\displaystyle d$ $\textstyle =$ $\displaystyle -{\textstyle{1\over 2}}\left\vert\begin{array}{ccc}{x_1}^2+{y_1}^...
...\  {x_2}^2+{y_2}^2 & y_2 & 1\\  {x_3}^2+{y_3}^2 & y_3 & 1\end{array}\right\vert$ (5)
$\displaystyle f$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\left\vert\begin{array}{ccc}{x_1}^2+{y_1}^2...
...\  {x_2}^2+{y_2}^2 & x_2 & 1\\  {x_3}^2+{y_3}^2 & x_3 & 1\end{array}\right\vert$ (6)
$\displaystyle g$ $\textstyle =$ $\displaystyle -\left\vert\begin{array}{ccc}{x_1}^2+{y_1}^2 & x_1 & y_1\\  {x_2}^2+{y_2}^2 & x_2 & y_2\\  {x_3}^2+{y_3}^2 & x_3 & y_3\end{array}\right\vert.$ (7)

Completing the Square gives
\begin{displaymath}
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
\end{displaymath} (8)

which is a Circle of the form
\begin{displaymath}
(x-x_0)^2+(y-y_0)^2=r^2,
\end{displaymath} (9)

with Circumcenter
$\displaystyle x_0$ $\textstyle =$ $\displaystyle -{d\over a}$ (10)
$\displaystyle y_0$ $\textstyle =$ $\displaystyle -{f\over a}$ (11)

and Circumradius
\begin{displaymath}
r=\sqrt{{f^2+d^2\over a^2}-{g\over a}}.
\end{displaymath} (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.



info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein
1999-05-26