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Euler's Distribution Theorem

For signed distances,

\begin{displaymath}
\overline{AB}\cdot\overline{CD}+\overline{AC}\cdot\overline{DB}+\overline{AD}\cdot\overline{BC}=0,
\end{displaymath}

since

\begin{displaymath}
(b-a)(d-c)+(c-a)(b-d)+(d-a)(c-b)=0.
\end{displaymath}


References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 3, 1929.




© 1996-9 Eric W. Weisstein
1999-05-25