Trilinears -1 + sec A/2 cos B/2 cos C/2 : -1 + sec B/2 cos C/2 cos A/2 : -1 + sec C/2 cos A/2 cos B/2
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)(-1 + sec A/2 cos B/2 cos C/2)
If a + b + c > 4R + r, where R and r denote the circumradius and inradius, respectively, then there exists a point X for which the perimeters of triangles XBC, XCA, XAB are equal. Veldkamp proved that X = X(175), and Yff, in unpublished notes, proved that X(175) is the center of the outer Soddy circle. See also the 1st and 2nd Eppstein points, X(481), X(482).
Clark Kimberling and R. W. Wagner, Problem E 3020 and Solution, American Mathematical Monthly 93 (1986) 650-652 [proposed 1983].
G. R. Veldkamp, "The isoperimetric point and the point(s) of equal detour," American Mathematical Monthly 92 (1985) 546-558.
X(175) lies on these lines: 1,7 8,1270 174,483 226,1131 490,664 651,1335
X(175) = X(8)-Ceva conjugate of X(176)
X(175) = X(664)-beth conjugate of X(175)
X(175) = {X(1),X(7)}-harmonic conjugate of X(176)
, select the center name from the list and, then, click on the vertices A, B and C successively.