Find if there is collision between two balls moving in a plane python

I have the following ball configuration:

P=(115,100)
T=(253,211)
C=(389,464)
r=60

The ball of center C and radius r follows a straight line from C to P.

In this trajectory there might be a ball T of radius r that causes a collision with ball C and deflects ball C from its course.

Ball of center T can be anywhere in the 2D plane except the blue region

Given C, T, P, and r, what is a clean and efficient way to detect whether or not ball T will interfer with ball C trajectory?

I would like to do this in python and other libraries can be used like numpy or pygame.

We can calculate distance D from point T to line CP.

Define vector

CP = P - C

in coordinates:

cpx = P.x - C.x
cpy = P.y - C.y

and normalize it

length = math.hypot(cpx, cpy)
ucpx = cpx / length
ucpy = cpy / length

Another vector

CT = T - C
ctx = T.x - C.x
cty = T.y - C.y

Distance using cross product

D = abs(CT x UCP)
D = abs(ctx*ucpy-cty*ucpx)

One more value - sign of dot product of vectors allows to know if T lies "behind" C

dtp =  (CT dot UCP)
dtp =  (ctx*ucpx+cty*ucpy)

Then compare D with radius sum (here doubled radius) and check sign of dot product

 if (D < 2*r) and (dtp > 0):
    collision occurs