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 Math Central Quandaries & Queries
 Question from George: I know the center location (x,y) of the circle, I know the radius of the circle, I know the location (X,Y) of one point on the circle, and I know the angle (in degrees not radians) between the known point location (X,Y) and an unknown point location (let's call it (A,B) for reference). What formula(s) can I use to find out the coordinate position of (A,B)?

George,

I would begin by translating the center to the origin so that the arithmetic is less complicated: subtract (x, y) from the given points to get (X', Y') = (X-x, Y-y) and (A',B') = (A-x, B-y). (After your computation you will add (x, y) to (A', B') to get the coordinates of (A,B).) Now find the angle the line joining the origin to (X',Y') makes with the x-axis. (The tangent of that angle is Y'/X', so the desired angle is arctan (Y'/X').) Add that angle to the given angle (taking care if either of the angles is below the x-axis and therefore negative) to get the angle the line from the origin to (A',B') makes with the x-axis; call it theta. Then A' = radius*cos(theta) and B' = radius*sin(theta).

Chris

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