Tixier Point
TixierPoint
The Tixier point X_(476) is the reflection of the focus of the Kiepert parabola (X_(110)) in the Euler line. It has equivalent trilinear center functions
X_(476) = [画像:(csc(B-C))/(1+2cos(2A))]
(1)
X_(476) = [画像:a/((b^2-c^2)[1+2cos(2A)]).]
![画像:a/((b^2-c^2)[1+2cos(2A)]).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TixierPoint/Inline8.svg){kind=link}
(2)
It lies on the circumcircle.
The distance between points X_(110) and X_(476) and the Euler line is given by
| d=(8Delta|(a^2-b^2)(b^2-c^2)(c^2-a^2)|)/((a^6-a^4b^2-a^2b^4+b^6-a^4c^2+3a^2b^2c^2-b^4c^2-a^2c^4-b^2c^4+c^6)^(3/2)), |
(3)
|
where Delta is the area of the reference triangle.
See also
Kiepert ParabolaExplore with Wolfram|Alpha
WolframAlpha
More things to try:
References
Kimberling, C. "Encyclopedia of Triangle Centers: X(476)=Tixier Point." https://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X476.Referenced on Wolfram|Alpha
Tixier PointCite this as:
Weisstein, Eric W. "Tixier Point." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TixierPoint.html