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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?

  • try giving a triangle coordinates $(x_1,y_1),(x_2,y_2),(x_3,y_3)$. The middle is the "average" of these points is the middle. Now try and find a condition where the middle does not lie within the bounds of the triangle – Henry Lee Oct 17 '18 at 19:07
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    In an obtuse triangle, the circumcentre is outside the triangle. – Angina Seng Oct 17 '18 at 19:11
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    @Henry: If you mean the centroid $(\frac{x_1+x_2+x_3}3,\frac{y_1+y_2+y_3}3)$, that's no good; the centroid is always inside the triangle. –  Oct 17 '18 at 19:12
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    For obtuse-angled triangles, the orthocentre is outside as well. – J.G. Oct 17 '18 at 19:57

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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".

dan_fulea
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