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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.

Euclid. The Thirteen Books of the Elements, 2nd ed. unabridged, Vol. 3: Books X-XIII New York: Dover, p. 36, 1956.

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




© 1996-9 Eric W. Weisstein
1999-05-26