Given a triangle $ABC$, let $A'$ be the middle point of $BC$, $B'$ the middle point of $AC$ and $C'$ the middle point of $AB$. It is well-known that the circumcenter of $ABC$ is the orthocenter of $A'B'C'$.
For acute triangles, orthocenter and circumcenter are inside the triangle, for right triangles both are at the border, and for obtuse triangles both are outside.
For me, this seems to imply that if the curcumcenter of a triangle $ABC$ is inside of the triangle, it must be inside of the triangle $A'B'C'$ (constructed as explained at the beggining of the question).
I haven't ever heard about this fact, which I find intuitive, since circumcenter shouldn't be too near from a particular vertex. Bit I'm a bit surprised of not having heard about it.
Is it true? Is it well-known?
You can easily see that $ABC$ and $A'B'C'$ are similar triangles. Hence, if the circumcenter of $ABC$ is inside $ABC$, then $ABC$ is acuteangle. This means that $A'B'C'$ is acuteangle, hence its orthocenter (which is the circumcenter of $ABC$) belongs to $A'B'C'$.
Roughly speaking, problems arise only for obtuse angle triangles, but if one between $ABC$ and $A'B'C'$ is obtuse, then so is the other one.