Friday, June 15, 2012

Problem 766: Triangle, Altitudes, Exradii, Three Exradius, Equal Harmonic Mean

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 766.

Online Geometry Problem 766: Triangle, Altitudes, Exradii, Three Exradius, Equal Harmonic Mean.

Posted by Antonio Gutierrez at 10:50 AM
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1 comment:

  1. Jacob Ha & W FungJune 16, 2012 at 3:28 AM

    Let Δ be area of triangle ABC, s = (a+b+c)/2

    Hence,
    1/(Ra) + 1/(Rb) = (s-a)/Δ + (s-b)/Δ =c/Δ = 2/(Hc)

    Symmetrically,
    1/(Ra) + 1/(Rb) =2/(Hc)
    1/(Rb) + 1/(Rc) =2/(Ha)
    1/(Rc) + 1/(Ra) =2/(Hb)

    Summing up,

    1/(Ra) + 1/(Rb) + 1/(Rc) = 1/(Ha) + 1/(Hb) + 1/(Hc)

    q.e.d

    ReplyDelete

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