TOPICS
Search

Clawson Point

ClawsonPoint

The extangents triangle is homothetic to the orthic triangle, and its homothetic center is known as the Clawson point, or sometimes the "crucial point." It has equivalent triangle center functions

alpha=tanA
(1)
alpha=sin(2B)+sin(2C)-sin(2A)
(2)

and is Kimberling center X_(19) (Kimberling 1998, p. 69).

Distances from the Clawson point to some other named triangle centers include

ClH=(8a^2b^2c^2IL|cosA||cosB||cosC|)/((a+b+c)(a^5-ba^4-ca^4+2bc^2a^2+2b^2ca^2-b^4a-c^4a+2b^2c^2a+b^5+c^5-bc^4-b^4c))
(3)
ClM=-(8abc(a+b+c)^2ILr^2)/((a^2-2ba-2ca+b^2+c^2-2bc)(a^5-ba^4-ca^4+2bc^2a^2+2b^2ca^2-b^4a-c^4a+2b^2c^2a+b^5+c^5-bc^4-b^4c))
(4)
ClSp=(2(a^3+ba^2+ca^2+b^2a+c^2a+2bca+b^3+c^3+bc^2+b^2c)ILr^2)/(a^5-ba^4-ca^4+2bc^2a^2+2b^2ca^2-b^4a-c^4a+2b^2c^2a+b^5+c^5-bc^4-b^4c),
(5)

where H is the orthocenter, M is the mittenpunkt, L is the de Longchamps point, and Sp is the Spieker center.

See also

Extangents Triangle, Homothetic Center, Orthic Triangle

Explore with Wolfram|Alpha

References

Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Encyclopedia of Triangle Centers: X(19)=Clawson Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X19.Lyness, R. and Veldkamp, G. R. Problem 682 and Solution. Crux Math. 9, 23-24, 1983.

Referenced on Wolfram|Alpha

Clawson Point

Cite this as:

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

Subject classifications