I am taking a course in multivariable calculus this year & I thought it would be a good idea to brush up plane and solid geometry.
I would like to prove that, for any given triangle, there is a unique circle that passes throught the three midpoints of sides, the feet of the altitudes, and the middle points on the segment joining the feet of the perpendicular and the orthocenter. Could you please give me any hints, that would lead to the correct proof?
For an example of the type of work I've done ... On some results related to triangle centers, I was able to prove that the centroid $G$ divides the line joining the orthocenter $H$ and the circumcenter $K$ in the ratio $2:1$. I wasn't sure how to prove that $H$, $G$, $K$ are collinear. I learnt from Kiselev's Geometry book about how to prove this result. The idea is that the circumcenter of the original triangle is the orthocenter of the medial triangle $H^\prime$. And the medial triangle reflected about the centroid $G$, followed by dilation of a factor $2:1$ results in the original triangle. Thus, the center $H^\prime$ moves to the point $H$. Both triangles share the centroid $G$. So, $H$, $G$, $K$ are collinear.