I have a question about a 3D check whether a point is inside a smallest possible sphere passing through a triangle $_▵ABC$ (also called equatorial sphere).
The 2D analogy is quite simple:
$P(px,py)$ is checked whether it is in the smallest circle possible passing through points $A(ax,ay)$ and $B(bx,by)$:
The quickest way to determine this is to check the sign of dot product $\vec{PA}·\vec{PB}$. If the result is negative, then the $cos(α)<0$, which means that $90<α<180$ and point is in the circle.
Now to the 3D example:
What is the quickest way to determine (preferably a similar way with vectors) if point $P(px,py,pz)$ is in the smallest sphere possible passing through points $A(ax,ay,az)$, $B(bx,by,bz)$, $C(cx,cy,cz)$?:
Thanks in advance for comments, hope I was clear enough.
Here is an outline of a method that should work.
The sphere is the smallest one containing points $A$, $B$ and $C$. The intersection of this sphere and the plane containing the points gives the circumcircle of the triangle $ABC$. In other words, the centre of the sphere is centre of the circumcircle of the triangle $ABC$.
Find the coordinates of the centre $O$ using the formula here.
Now find a point at the opposite end of a diameter for the sphere starting at, say, $A$. It's coordinates can be obtained by adding $2 \overrightarrow{AO}$ to the coordinates of $A$. Call this new point $A'$.
Now you can repeat your 2-D test using the points $A$, $A'$ and $P$.