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Viviani's Theorem

For a point $P$ inside an Equilateral Triangle $\Delta ABC$, the sum of the perpendiculars $p_i$ from $P$ to the sides of the Triangle is equal to the Altitude $h$. This result is simply proved as follows,

\Delta ABC=\Delta PBC+\Delta PCA+\Delta PAB.
\end{displaymath} (1)

With $s$ the side length,
{\textstyle{1\over 2}}sh={\textstyle{1\over 2}}sp_a+{\textstyle{1\over 2}}sp_b+{\textstyle{1\over 2}}sp_c,
\end{displaymath} (2)

\end{displaymath} (3)

See also Altitude, Equilateral Triangle

© 1996-9 Eric W. Weisstein