Monday, April 19, 2010

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

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.

Problem 435: Triangle, Incircle, Inradius, Congruent circles, tangent, Measurement.
See also:
Distance between the Incenter and the Centroid
Level: High School, SAT Prep, College geometry

Posted by Antonio Gutierrez at 3:01 PM
Labels: , , , , , ,

1 comment:

  1. AnonymousApril 22, 2010 at 5:10 AM

    we have: p=(a+b+c)/2
    Heron'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!
    .-.

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