Problem statement
Let $\triangle ABC$ a triangle and $M$ a point inside it. Let $\mathcal{C_c}$ be the circumcircle of $\triangle MAB$ and $\mathcal{C_a}$ and $\mathcal{C_b}$ similarly defined. Note $R_a$, $R_b$ and $R_c$ the radii of these circles. Prove that: $$R_a + R_b + R_c \geq MA + MB + MC$$
Figure (made using GeoGebra Geometry)
Attempt (basic observations)
Firstly, I denoted the circumcenters $O_a$, $O_b$ and $O_c$, as in the figure. The statement rewrites: $$MO_a + MO_b + MO_c \geq MA + MB + MC$$
Secondly, I have tried to find the solution for the equality case. I observed that, if $M := O$, $MA = MB = MC$, but still nothing to relate with $MO_a$, $MO_b$, $MO_c$. But I think this may be the point I'm looking for.
Background and geometric tools that I am able to use
Synthetic geometry (Euclidean), I am also able to use vectors and analytic geometry (only Cartesian formulae).