Let $C_1$ and $C_2$ be two concentric circles in the plane with radii $R$ and $3R$ respectively. Show that the orthocentre of any triangle inscribed in circle $C_1$ lies in the interior of circle $C_2$. Conversely, show that also every point in the interior of $C_2$ is the orthocentre of some triangle inscribed in $C_1$.
For equilateral triangle circumcenter, centroid , orthocenter coincides ,for right triangles orthocenter is on circum circle ,for obtuse triangles is between r to 3r and for other triangles it is inside circumcircle. Points circumcenter(c), centroid(g) and orthocenter(h) are collinear g divides ch in 1:2 so when g lies on circumcircle g is 3r from c. But when it comes to converse we see points between circle(3r) - circle(r) are swept by rotating that collinear line by symmetry which is allowed as by rotating line triangli will rotate on center will not violate requirements