Is the line joining the circumcentre and the foot of the median perpendicular to that side?

Question

This is the picture. Here, is the line SP perpendicular on the side BC?

Answer 1

Yes. By definition.

The circumcenter is intersection of the perpendicular bisectors. That means it's perpendicular and a bisector. A median is a bisector (It needn't be perpendicular) but that means that the median and the perpendicular bisector start off at the same place (the midpoint of that side of the triangle). They start off on the same foot!

Answer 2

Indeed, right triangles $SPB,SPC$ are congruent.. by definition of any circumcenter and its construction method. $SP$ not only bisects but is perpendicular to side $BC$.

Answer 3

The circumcenter $S$ is the point such that $AS=BS=CS=R$. It implies $\Delta BCS$ is an isosceles triangle. The altitude $SP$ is a bisector as well as a median of $\Delta BCS$, hence $P$ is also the foot of the median of $\Delta ABC$.