KöMaL 2017 B4856

Point $P$ is said to be a centre of convexity of a point set $\mathcal{H}$ if $\mathcal{H}\cup H_P'$ is convex, where $H_P'$ is the reflection of the point set $\mathcal{H}$ in the point $P$. Show that $a)$ every convex quadrilateral of the plane has three distinct non-collinear centres of convexity; $b)$ a tetrahedron has no centre of convexity.