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Math Help - Incricles and Excircles

  1. #1
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    Incricles and Excircles

    If I have r as an inradius of a triangle and ra, rb, rc are the three radii of the 3 excircles of that triangle how can I show that:

    1/ra+1/rb+1/rc=1/r?
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  2. #2
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    Quote Originally Posted by TheNcredibleHulk View Post
    If I have r as an inradius of a triangle and ra, rb, rc are the three radii of the 3 excircles of that triangle how can I show that:

    1/ra+1/rb+1/rc=1/r?
    Start here, with the formulas r = A/s, r_a = A/(s-a), r_b = A/(s-b), r_c = A/(s-c), where A is the area of the triangle and s = \tfrac12(a+b+c).
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    Im still not getting this.
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  4. #4
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    Quote Originally Posted by Opalg View Post
    Start here, with the formulas r = A/s, r_a = A/(s-a), r_b = A/(s-b), r_c = A/(s-c), where A is the area of the triangle and s = \tfrac12(a+b+c).
    If you are willing to accept those formulas, then the calculation goes like this:

    \frac1{r_a}\, +\, \frac1{r_b}\, +\, \frac1{r_c} = \frac{(s-a)+(s-b)+(s-c)}A = \frac{3s-(a+b+c)}A = \frac{3s-2s}A = \frac sA = \frac1r.
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