The third Lemoine circle, a term coined here for the first time, is the circumcircle of the Lemoine triangle. It has center function α = (f(a, b, c))/a, where f(a, b, c) is a 10th-order polynomial, which is not a Kimberling center and radius R = sqrt((g(a, b, c) g(b, c, a) g(c, a, b))/((-a + b + c)(a + b - c)(a - b + c)(a + b + c)))/(2(4a^2 + b^2 + c^2)(a^2 + 4b^2 + c^2)(a^2 + b^2 + 4c^2)), where g(a, b, c) = - 3a^6 + b^2 a^4 + 3c^2 a^4 + 5b^4 a^2 + 3c^4 a^2 + 23b^2 c^2 a^2 + b^6 - 3c^6 + b^2 c^4 + 5b^4 c^2.
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