## 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.

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

Level: High School, SAT Prep, College geometry

#### 4 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

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.

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

4. Draw a line through point D parallel too the line AC. The line meets the line AB in M and BC in N.

The line BH meets the line MN in the point Q.

BQ=h-g

Draw an altitude from point M onto the line BN. The altitude meets the line BN on the point P.Because the triangle BMN is equilateral we have:

MP=BQ=h-g

Also from problem 223 we have:

MP=e+f

Therefore

e+f=h-g
e+f+g=h