Geometry Problem

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

Click the figure below to see the complete problem 766.

## Friday, June 15, 2012

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

Labels:
altitude,
exradius,
harmonic mean,
triangle

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

ReplyDeleteHence,

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