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Second Eppstein Point

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SecondEppsteinPoint

Let I_A, I_B, and I_C be the vertices of the inner 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_AI_A, E_BI_B, and E_CI_C concur at a point known as the second Eppstein point (Kimberling), denoted M by Eppstein (2001). Although Eppstein (2001) actually cited (Oldknow 1996), he missed the fact that M is equivalent to the outer Oldknow point defined by Oldknow (1996).

SecondEppsteinPointOldknow

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

The second Eppstein point has equivalent triangle center functions

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

where Delta is the area of DeltaABC, and is Kimberling center X_(482).

See also

First 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(482)=2nd Eppstein Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X482.Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319-329, 1996.

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Second Eppstein Point

Cite this as:

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

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