Prove the the orthocentre of a triangle which is incribed in a circle is inside of the concentric circle of 3 times radius.

Question

The question is : Let C and C' be two concentric circles in the plane with radii R & 3R respectively. Show that the orthocentre of any triangle inscribed in circle C lies in the interior of circle C'. It is a good problem I believe . In observation we can easily see that It is true because C' is a very big circle but in which way we can solve it I need some help.. Thank you

Answer 1

It is enough to exploit Euler's theorem. We know that $O,G,H$ (circumcenter, centroid, orthocenter) are always collinear and $HO=3 OG$. Let $O$ be the circumcenter of our triangle, i.e. the centre of $C$. The centroid of our triangle lies inside such a triangle, hence in the interior of $C$.
It follows that $H$ lies in the interior of $3C=C'$.