A circle with center at O is divided into n equal arcs, n≥ 2, by the
points A₁A₂ A₃....A_{n} Find the sum of radius vectors i OA_{i}
where
, i=1,2,...,n.
Let's use the circle whose radius is in the complex plane. Assume that .
Your points are the roots of the unity. Let be a primitive root of the unity (i.e. and ).
Then your sum is
If you want to generalize this to your problem, just apply a homothety and a rotation to the circle we used to obtain yours.
Hello, makenqau!
A circle with center at is divided into equal arcs,
by the points:
Find the sum of radius vectors: .
This is a classic problem . . .
I recall that the sum is zero, but the proof eludes me.
With that in mind, you may be able to construct your own proof.