Doesn’t a triangle center have to be inside of the triangle? A ‘Tarry point’, which is on the circumcircle of the given triangle, is described as a ‘triangle center’ by Wikipedia in its article on the Steiner point:
“The Tarry point is a triangle center and it is designated as the center X(98) in Encyclopedia of Triangle Centers.”
(https://en.wikipedia.org/wiki/Steiner_point_(triangle))
So, how is it that a ‘triangle center’ can be outside of the interior of the triangle in question?
 A: The Encyclopedia of Triangle Centers, ETC for short collects points related to a given triangle. It is a "destructive approach" from the point of view of the synthetic geometry, and the main idea is that starting from a triangle $\Delta ABC$, considered "fixed" with side lengths $a,b,c$ each point $P$ in the plane of the triangle has barycentric coordinates $(x:y:z)$ w.r.t. the triangle, so that formally
$$
P = \frac 1{x+y+z}(xA+yB+zC)\ ,
$$
(to give a sense to the above, replace each point by its affix, or use a vectorial interpretation $OP=\frac 1{x+y+z}(x\cdot OA+y\cdot OB+z\cdot OC)$ with vectors $OP$, $OA$, $OB$, $OC$ in the formula). 
Using barycentric coordinates to show properties of points constructed in a triangle is often a good way to proceed. (It has a taste of analytic geometry, but we are not using cartesian coordinates, but barycentric coordinated, directly related to the triangle we start with, and a taste of algebraic geometry, since soon we write rational expressions in $a,b,c$.)
Now for each rational function $f$ of three variables, which usually satisfies
$$
f(a,b,c)=f(a,c,b)
$$
we may define the point with barycentric coordinates $(x:y:z)$ given by
$$
\begin{aligned}
x &= f(a,b,c)\ ,\\
y &= f(b,c,a)\ ,\\
z &= f(c,a,b)\ .
\end{aligned}
$$
Lines and circles in the world using barycentric coordinates are described by algebraic equations, see for instance the excellent compact quide
https://web.evanchen.cc/handouts/bary/bary-short.pdf
and geometrical constructions lead to building and solving systems of equations (when possible). So we have formulas for "simple points" (like the centroid), e.g. $X(2)=(1:1:1)$, and we have formulas for "complicated points" like $X(98)=1/(b^4+c^4-a^2(b^2+c^2))$.
The usage of the word "center" in ETC is thus rather a synonym for a "remarcable point".   
