Is at least 1 of 4 non-concyclic points contained in the circle through the other 3?

Among all four circles you can make, take the one with the biggest radius.

Say it goes through $ABC$ and $D$ is fourth point and let $A$ and $D$ be on a different sides of line $BC$. Suppose $D$ is outside of the circle and let $AD$ cut the circle at $E$. Draw another circle thorugh $ACD$ with (red) radius $R'$. Then if $CD > CE$ and thus $$ \color{red}{ 2R' ={CD \over \sin \phi}} >{CE \over \sin \phi} =2R$$ a contradiction. So $CE>CD$ and thus $D$ is in the (black circe).