@Markiyan Hirnyk   My code has an important advantage over  dsolve/events  method, because it uses the explicit equation of the trajectory. For example, try to find the trajectory by using  dsolve/events  in the time range  t=10000 .. 10020, if the initial conditions are given for t=0 .

@Markiyan Hirnyk Of course, some variants are only for presentation of result. For the calculations the form  A  can be used. 

@Carl Love  Because the chairs with numbers  1  and  8  are next to each other.

@Dira 

PartialSum1:=proc(n::posint)

local S, k;

S:=0:

for k to n do

S:=S+4*(-1)^(k-1)/(2*k-1);

od;

evalf(S);

end proc:

@Timm  

restart;

b:=c-a:

[a+b+2, a+2+b, b+2+a];

               [c+2, c+2, c+2]

@J4James   The same can be done in one line without calling  plots  package:

restart;

plot([[seq([n, sin((1/10)*n)], n=0..30)], [seq([n, cos((1/10)*n)], n=0..30)]], color=[red, green], style=[line, point]);

@litun    mul(s-a[i], i={$1..100} minus {2});

@Markiyan Hirnyk   I did not notice the obvious: with respect to  f(x)  we have the usual quadratic equation.

@Carl Love  Thanks for the useful information on more intelligent   i  &^ n mod m  command .

@acer  Thank you very much for the detailed analysis of the situation!

@Carl Love   The center of the described circle can be not only inside but also outside the triangle. You have considered only the first case.

@Markiyan Hirnyk   I observed that  if ahead as a factor there is the empty symbol, then it "pulls" a forward all numeric constants, for example

``*sin(x)*sqrt(2)+sqrt(2)*cos(x);

@Markiyan Hirnyk   My output in Classic M 12 .

@Markiyan Hirnyk  Take a closer look at my calculations. I do not see a significant difference:

evalf([F(0), F(1), F(2)]);

   [0.03556431828, 0.03160918497, 0.02152263368]

@Markiyan Hirnyk  I don't understand the meaning of your question.