How to calculate point on circle from angle between middle and other point on the circle

Question

I have circle with diameter 256px, middle point is on xy[128,128]. I have first point on circle, for example X=0,Y=128. Coordinates origin is in left top corner. Line between first point and middle, together with angle on this line (starting from middle) creates triangle which intersects circle in third point. How can I calculate this point?

Input variables are :

1. point xy on circle
2. middle point
3. angle

expected output is x1,y1

Answer 1

Formulas for rotation of initial point around center point by angle:

 x1 = middle.x + (x - middle.x) * cos(angle) - (y - middle.y) * sin(angle)
 y1 = middle.y + (x - middle.x) * sin(angle) + (y - middle.y) * cos(angle)

(this is affine transform - combination of translation middle to origin, rotation about origin and backward translation)

Don't forget to make cos and sin argument in radians rather than degrees like this:

 cos(angle * M_PI / 180)