For a point 
inside an equilateral triangle 
, the sum of the perpendiculars 
from 
to the sides of the triangle
is equal to the altitude 
. This result is simply proved as follows,
 |
(1)
|
With 
the side length,
 |
(2)
|
so
 |
(3)
|
Viviani's Theorem
See also
Altitude, Equilateral TriangleExplore with Wolfram|Alpha
References
Bogomolny, A. "Viviani's Theorem." http://www.cut-the-knot.org/Curriculum/Geometry/Viviani.shtml.Bogomolny, A. "Viviani's Theorem: Proof Without Words." http://www.cut-the-knot.org/Curriculum/Geometry/VivianiPWW.shtml.Referenced on Wolfram|Alpha
Viviani's TheoremCite this as:
Weisstein, Eric W. "Viviani's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VivianisTheorem.html