## Monday, January 12, 2009

### Elearn Geometry Problem 222: Viviani Theorem, Exterior point

Equilateral Triangle, Exterior Point
For a given point D outside an equilateral triangle ABC, the sum of the distances DE and DF minus DG from the point D to the sides BC, AB and AC respectively is equal to the altitude of the triangle.

See complete Problem 222 at:
gogeometry.com/problem/p222_viviani_theorem_equilateral_triangle_exterior.htm

Level: High School, SAT Prep, College geometry

1. Join D to A & C. Now Tr. ABC = Area DABC - Tr ADC = Tr.DAB + Tr. DAC -Tr.ADC Hence,
Tr.ABC = Tr.DAB + Tr. DAC -Tr.ADC or
AC*h/2 = AB*f/2 + AC*e/2 - AC*g/2 or
h = f + e - g
QED
Ajit: ajitathle@gmail.com

2. Draw a line through the point D parallel to the line AC. Extend BF to meet the line parallel to AC in the point P. And also extend the line BE to meet the line parallel to AC in the point Q.

Extend the line BH to meet the lin PQ in the point N. Then :

BN=h+g

Draw a altitude from the point P onto the line BQ. Because the triangle BPQ is equilateral.

PM=BN=h+g

From problem 223 we also know that :

PM=e+f

Therefore :

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