Wednesday, January 14, 2009

Elearn Geometry Problem 225: Viviani Theorem Extension

Regular Polygon, Apothem, Distance
In a regular n-sided polygon, the sum of the perpendicular distances from an interior point to the n sides being n times the apothem of the polygon.

 Geometry Problem 225. Viviani Theorem Extension
See complete Problem 225 at:

Level: High School, SAT Prep, College geometry


  1. One can prove that, in a equiangled polygon, the sum of the distances of an arbitrary inner point to the sides is constant =]

  2. (1)let the no. of sides be n.
    (2)let the length of each side be y.
    (3)therefore the area of the given polygon is 1/2ap.y.n
    (4)from point P, we can do tringulations (on the sides)
    (5)We note that sum of area of those triangls=area by 1/2 ap.y.n method =(h1+h2+h3+....+hn).y.1/2
    (6)then, 1/2& y get cancelled & we get ap.n = h1+h2+h3+...+hn
    (7)hence the proof!!