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Viviani's Theorem

Viviani's theorem states that for a point P inside an equilateral triangle DeltaABC, 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,

DeltaABC=DeltaPBC+DeltaPCA+DeltaPAB.
(1)

With s the side length,

1/2sh=1/2sp_a+1/2sp_b+1/2sp_c,
(2)

so

h=p_a+p_b+p_c.
(3)

See also

Altitude, Equilateral Triangle

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References

Bogomolny, A. "Viviani's Theorem." https://www.cut-the-knot.org/Curriculum/Geometry/Viviani.shtml.Bogomolny, A. "Viviani's Theorem: Proof without Words." https://www.cut-the-knot.org/Curriculum/Geometry/VivianiPWW.shtml.

Referenced on Wolfram|Alpha

Viviani's Theorem

Cite this as:

Weisstein, Eric W. "Viviani's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VivianisTheorem.html

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