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Equal Detour Point


The center of an inner Soddy circle. It has equivalent triangle center functions

alpha=1+(2Delta)/(a(b+c-a))
(1)
alpha=sec(1/2A)cos(1/2B)cos(1/2C)+1,
(2)

where Delta is the area of the triangle.

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

 |AY|+|YB|-|AB|
(3)

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 >2sin^(-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.


See also

Isoperimetric Point, Soddy Circles

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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://faculty.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.

Referenced on Wolfram|Alpha

Equal Detour Point

Cite this as:

Weisstein, Eric W. "Equal Detour Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EqualDetourPoint.html

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