Proposed Problem

Prove that the distance between the incenter I and the centroid G of a triangle ABC is:

Click the figure below to see the complete illustration.

See also:

Distance between the Incenter and the Centroid

Level: High School, SAT Prep, College geometry

## Monday, April 19, 2010

### Problem 690: Distance between the Incenter and the Centroid of a Triangle

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we have: p=(a+b+c)/2

ReplyDeleteHeron's formula S²=p(p-a)(p-b)(p-c)

the cosine law cosA=(b²+c²-a²)/(2bc)

S=pr

with origin A;x-axis AC

I(p-a;r)=(p-a;S/p)

let be :D the midpoint of AC;BE the altitude

GF//BE

the triangles DGF and DBE are similar

yG=BE/3=2S/3b

xG=AF=AD+DF

AD=b/2

DF=DE/3=(AE-AD)/3

AE=ABcosA=(b²+c²-a²)/2b

xG=(3b²+c²-a²)/(6b)

d²=(xI-xG)²+(yI-yG)²

and a little algebra gives the result!

.-.