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Noam Elkies proved that if $x,y,z$ are positive numbers, then there is a unique point $P$ inside $ABC$ such that the inradii $r_a,r_b,r_c$ of the triangles $BPC, CPA, APB,$ respectively, satisfy

$$ r_a : r_b : r_c = x : y : z.$$

Can someone find barycentric coordinates for Elkies points?

Elkies's proof appears in Math. Mag. 60 (1987), 117. I'd especially like to have such coordinates when $x : y : z$ are barycentric coordinates of a well-known point, such as the incenter, centroid, circumcenter, Fermat point, or a Hofstadter point.

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  • $\begingroup$ So I'm guessing you would like to get the barycentric coordinates of $P$ in terms of $r_a:r_b:r_c$ and the side lengths $a,b,c?$ $\endgroup$
    – Aaron Meyerowitz
    Commented Sep 21, 2018 at 5:20
  • $\begingroup$ Aaron, yes, or equivalently, in terms of $r_a, r_b, r_c$ and the angles $A, B, C$, or all nine: $r_a, r_b, r_c, a, b, c, A, B, C$. $\endgroup$
    – Clark Kimberling
    Commented Sep 21, 2018 at 14:12

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