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First Eppstein Point

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FirstEppsteinPoint

Let J_A, J_B, and J_C be the vertices of the outer Soddy triangle, and also let E_A, E_B, and E_C be the pairwise contact points of the three tangent circles. Then the lines E_AJ_A, E_BJ_B, and E_CJ_C concur at a point known as the first Eppstein point Ep_1 (Kimberling), denoted M^' by Eppstein (2001). Although Eppstein (2001) actually cited (Oldknow 1996), he missed the fact that M^' is equivalent to the inner Oldknow point defined by Oldknow (1996).

FirstEppsteinPointOldknow

The first Eppstein point (originally called the inner Oldknow point) is also the perspectors of a given triangle DeltaABC and the tangential triangles of its outer Soddy triangle (Oldknow 1996).

The first Eppstein point has equivalent triangle center functions

alpha_(481)=1-2cos(1/2B)cos(1/2C)sec(1/2A)
(1)
alpha_(481)=1-(4Delta)/(a(-a+b+c)),
(2)

where Delta is the area of DeltaABC, and is Kimberling center X_(481).

See also

Second Eppstein Point, Soddy Circles, Soddy Triangles

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References

Danneels, E. "The Eppstein Centers and the Kenmotu Points." Forum Geom. 5, 173-180, 2005. http://forumgeom.fau.edu/FG2005volume5/FG200523index.html.Eppstein, D. "Tangent Spheres and Triangle Centers." Amer. Math. Monthly 108, 63-66, 2001.Kimberling, C. "Encyclopedia of Triangle Centers: X(481)=1st Eppstein Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X481.Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319-329, 1996.

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First Eppstein Point

Cite this as:

Weisstein, Eric W. "First Eppstein Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FirstEppsteinPoint.html

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