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

## Monday, January 12, 2009

### Elearn Geometry Problem 221: Viviani Theorem

Labels:
distance,
equilateral,
interior point,
perpendicular,
triangle,
Viviani theorem

Subscribe to:
Post Comments (Atom)

Tr.ABC = Tr.BDA + Tr.ADC + Tr.CDB

ReplyDeleteAC*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

area of ABC = (1/2)*h*AC

ReplyDeletealso 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.

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

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

ReplyDeleteThe 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