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Orthic Triangle

\begin{figure}\begin{center}\BoxedEPSF{orthic_triangle.epsf scaled 1000}\end{center}\end{figure}

Given a Triangle $\Delta A_1A_2A_3$, the Triangle $\Delta H_1H_2H_3$ with Vertices at the feet of the Altitudes (perpendiculars from a point to the sides) is called the orthic triangle. The three lines $A_iH_i$ are Concurrent at the Orthocenter $H$ of $\Delta A_1A_2A_3$.


The centroid of the orthic triangle has Triangle Center Function

\begin{displaymath}
\alpha=a^2\cos(B-C)
\end{displaymath}

(Casey 1893, Kimberling 1994). The Orthocenter of the orthic triangle has Triangle Center Function

\begin{displaymath}
\alpha=\cos(2A)\cos(B-C)
\end{displaymath}

(Casey 1893, Kimberling 1994). The Symmedian Point of the orthic triangle has Triangle Center Function

\begin{displaymath}
\alpha=\tan A\cos(B-C)
\end{displaymath}

(Casey 1893, Kimberling 1994).

See also Altitude, Fagnano's Problem, Orthocenter, Pedal Triangle, Schwarz's Triangle Problem, Symmedian Point


References

Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 9, 1893.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9 and 16-18, 1967.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.




© 1996-9 Eric W. Weisstein
1999-05-26