Question

I have the following setting:

I know P1, P2 and the angle alpha, now how do i calculate the coordinates of P3? (Note: P3 is on the same circle with origin P1 and radius P1P2)

The blue lines indicate the coordinate system

Answer 1

The formula stated above from Wikipedia is usable here to rotate the vector P1->P2 (V12).

V12 = [0, -100]

When rotated (beware of α is -30 degrees in your drawing) the vector P1->P3 becomes

x' = V12(x)*cos(α) - V12(y)*sin(α) = 0*cos(-30) - (-100)*sin(-30) = -50
y' = V12(x)*sin(α) + V12(y)*cos(α) = 0*sin(-30) + (-100)*cos(-30) = -86.6

When translated with the point P1 the coordinates for P3 becomes

[x, y] = [-50+150, -86.6+210] = [100, 123.4]