Equal Detour Point

The center of an outer Soddy Circle. It has Triangle Center Function

$$\alpha=1+{2\Delta\over a(b+c-a)} = \sec({\textstyle{1\over 2}}A)\cos({\textstyle{1\over 2}}B)\cos({\textstyle{1\over 2}}C)+1.$$

Given a point $Y$ not between $A$ and $B$, a detour of length

$$\vert AY\vert+\vert YB\vert-\vert AB\vert$$

is made walking from $A$ to $B$ via $Y$, the point is of equal detour if the three detours from one side to another via $Y$ are equal. If $ABC$ has no Angle $>2\sin^{-1}(4/5)$, then the point given by the above Trilinear Coordinates is the unique equal detour point. Otherwise, the Isoperimetric Point is also equal detour.

References

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.

Kimberling, C. ``Isoperimetric Point and Equal Detour Point.'' http://cedar.evansville.edu/~ck6/tcenters/recent/isoper.html.

Veldkamp, G. R. ``The Isoperimetric Point and the Point(s) of Equal Detour.'' Amer. Math. Monthly 92, 546-558, 1985.