Monday, January 12, 2009

Elearn Geometry Problem 221: Viviani Theorem

Equilateral Triangle, Interior Point
In an equilateral triangle, prove that the sum of the distances from any interior point to the sides is equal to the altitude of the triangle.


 Geometry Problem 221. Viviani Theorem
See complete Problem 221 at:
gogeometry.com/problem/p221_viviani_theorem_equilateral_triangle.htm

Level: High School, SAT Prep, College geometry

3 comments:

  1. Tr.ABC = Tr.BDA + Tr.ADC + Tr.CDB
    AC*h/2 = BA*f/2 + AC*g/2 + CB*e/2
    But AC=CB=BA since ABC is equilateral. Hence h=f+g+e
    QED
    Ajit: ajitathle@gmail.com

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  2. area of ABC = (1/2)*h*AC
    also area of ABC = area of ADC + area of BDA + area of BDC = (1/2)*g*AC + (1/2)*f*AB + (1/2)*e*BC.
    which implies that h*AC = g*AC + f*AB + e*BC
    but AB = AC = BC since ABC is equilateral.
    therefore h = e + f + g.

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  3. http://www.youtube.com/watch?v=0AOqIfTxCDg

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