Equidistant Triangle Centers (take 2)

The Nagel point Na has barycentric coordinates $s-a:s-b:s-c$ . As the isotomic conjugate of Na, the Gergonne point Ge has barycentric coordinates $\tfrac{1}{s-a}:\tfrac{1}{s-b}:\tfrac{1}{s-c}$ . The centroid G has barycentric coordinates $1:1:1$ .

The distance between points P and Q with normalized barycentric coordinates $p_1:p_2:p_3$ and $q_1:q_2:q_3$ (so that $p_1+p_2+p_3=q_1+q_2+q_3=1$ ) is given by the formula $PQ^2 = -a^2(p_2-q_2)(p_3-q_3) - b^2(p_1-q_1)(p_3-q_3) - c^2(p_1-q_1)(p_2-q_2)$ Normalizing the coordinates for Ge, Na and G, we calculate that $GeG^2 - NaG^2 \; = \; \frac{8X}{(a+b+c)(a^2+b^2+c^2 - 2ab - 2ac - 2bc)^2}$ where $X \; = \; \begin{array}{l} (a^6b + ab^6 + a^6c + ac^6 + b^6c + bc^6) - 3(a^5b^2 + a^3b^5 + a^5c^2 + a^2c^5 + b^5c^2 + b^2c^5) + 2(a^4b^3 + a^3b^4 + a^4c^3 + a^3c^4 + b^4c^3 + b^3c^4) \\{} - 4abc(a^4 + b^4 + c^4) + 7abc(a^3b + a^3c + ab^3 + b^3c + ac^3 + bc^3) - 8abc(a^2b^2 + a^2c^2 + b^2c^2) - 2a^2b^2c^2(a + b + c) \end{array}$ We want to find triangles $ABC$ for which $X=0$ . With $b=5\,,\,c=9$ this equation becomes $0 \; =\; 2 (80640 + 13568 a + 12894 a^2 - 13919 a^3 + 3059 a^4 - 249 a^5 + 7 a^6)$ which has two positive real roots, $a =4.16864$ and $a = 6.70671$ . Since $b = 5 < a < 9 = c$ , we deduce that $a = \boxed{6.70671}$ .