When point is inside of circle Let we have points $A_1,A_2,A_3,A_4$.In time $t$ point $A_i$ has coordinates $(x_i,y_i) + (v_{xi},v_{yi}) * t$, all parametrs are given. Describe algorithm to find all $t$ when point $A_4$ inside circle circumscribed around $A_1,A_2,A_3$ , or to find first moment when this happens.

First,
find the center and radius
of the circle.
Ways to do this
are shown here:
Get the equation of a circle when given 3 points
Then,
find the times
that the line passes through
the circle.
Using vectors,
if the center is at $C$,
the radius is $r$,
and the line is
$A+Bt$,
you want
$|A+Bt-C|
=r$
or
$|A+Bt-C|^2
=r^2$
or
$\begin{array}\\
r^2
&=\sum (A_i-C_i+tB_i)^2\\
&=\sum (D_i+tB_i)^2
\qquad D_i = A_i-C_i\\
&=\sum (D_i^2+2tD_iB_i+t^2B_i^2)\\
&=\sum D_i^2+2t\sum D_iB_i+t^2\sum B_i^2
\end{array}
$
This is a quadratic in $t$
and can be solved
by the standard formula.
The values of $t$
tell when the line
intersects the circle.
In terms of dot products,
the equation for $t$ is
$r^2
=D\cdot D+2tD\cdot B+t^2B\cdot B
$.
