Angle Doubling, Tripling, and Halving Construction

Draw a geometry construction centered at a fixed point A with a circle. Mark a base angle by choosing two points on the circle that determine rays from A; use another point O on that circle to define an angle region together with the point P. Construct an auxiliary circle with center F and place a point G on it, then draw the line through F–G and create perpendicular lines from F to two chosen directions so they intersect the auxiliary circle at several points (like H, I, J). Connect the relevant intersection points with short segments to serve as angle references. Rotate the ray around F by a fixed angle (e.g., 30°) to obtain two rotated rays, forming angles at F corresponding to doubled/tripled and halved versions. Finally, on the original circle, draw a circular arc between two auxiliary points Q and R to visualize the angle transformation.