Orthogonal Circles

$\begin{figure}\begin{center}\BoxedEPSF{OrthogonalCircles.epsf scaled 500}\end{center}\end{figure}$

Orthogonal circles are Orthogonal Curves, i.e., they cut one another at Right Angles. Two Circles with equations

$\begin{displaymath} x^2+y^2+2gx+2fy+c=0 \end{displaymath}$

(1)

$\begin{displaymath} x^2+y^2+2g'x+2f'y+c'=0 \end{displaymath}$

(2)

are orthogonal if

$\begin{displaymath} 2gg'+2ff'=c+c'. \end{displaymath}$

(3)

$\begin{figure}\begin{center}\BoxedEPSF{OrthogonalCirclesTheorem.epsf scaled 700}\end{center}\end{figure}$

A theorem of Euclid states that, for the orthogonal circles in the above diagram,

$\begin{displaymath} OP\times OQ=OT^2 \end{displaymath}$

(4)

(Dixon 1991, p. 65).

References

Dixon, R. Mathographics. New York: Dover, pp. 65-66, 1991.

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