## X(176) (EQUAL DETOUR POINT)

 Interactive Applet

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 Information from Kimberling's Encyclopedia of Triangle Centers

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 X is a point not between A and B, we make a detour of magnitude |AX| + |XB| - |AB| if we walk from A to B via X; then a point has the equal detour property if the magnitudes of the three detours, A to B via X, B to C via X, and C to A via X, are equal; X(176) is the only such point unless ABC has an angle greater than 2*arcsin(4/5), and then X(175) also has the equal detour property. Yff found that X(176) is also is the center of the inner Soddy circle. The following construction was found by Elkies: call two circles within ABC companion circles if they are the incircles of two triangles formed by dividing ABC into two smaller triangles by passing a line through one of the vertices and some point on the opposite side; chain of circles O(1), O(2), ... such that O(n),O(n+1) are companion incircles for every n consists of at most six distinct circles; there is a unique chain consisting of only three distinct circles; and for this chain, the three subdividing lines concur in X(176).

G. R. Veldkamp, "The isoperimetric point and the point(s) of equal detour," American Mathematical Monthly 92 (1985) 546-558.

Noam D. Elkies and Jiro Fukuta, Problem E 3236 and Solution, American Mathematical Monthly 97 (1990) 529-531 [proposed 1987].

X(176) lies on these lines: 1,7    8,1271    174,1143    226,1132    489,664    651,1124

X(176) = X(8)-Ceva conjugate of X(175)
X(176) = X(664)-beth conjugate of X(176)
X(176) = {X(1),X(7)}-harmonic conjugate of X(175)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.