XVII OM - I - Problem 4

On a plane, a circle and a point $ M $ are given. Find points $ A $ and $ B $ on the circle such that the segment $ AB $ has a given length $ d $, and the angle $ AMB $ is equal to a given angle $ \alpha $.
Suppose $ A $ and $ B $ are points on a given circle satisfying the conditions $ AB = d $, $ \measuredangle AMB = \alpha $ (Fig. 5).
Let us choose any chord $ CD $ of length $ d $ in the given circle. Rotate the triangle $ AMB $ around the center $ O $ of the given circle by the angle $ HOK $, where $ H $ and $ K $ denote the midpoints of the chords $ AB $ and $ CD $. After the rotation, point $ A $ will be at point $ C $, and point $ B $ at point $ D $ or vice versa; point $ M $ will move to some point $ N $. Point $ N $ a) lies on the circle with center $ O $ and radius $ OM $, b) lies on such an arc of the circle whose endpoints are points $ C $ and $ D $ and which contains the inscribed angle $ \alpha $. Conversely, if we determine a point $ N $ satisfying the above conditions a) and b), then by rotating the triangle $ CND $ around $ O $ by the angle $ NOM $, we obtain the triangle $ AMB $, whose base is the desired segment. Since there are two arcs with endpoints $ C $, $ D $ containing the inscribed angle $ \alpha $, the number of solutions will be $ 4 $, $ 3 $, $ 2 $, $ 1 $, or $ 0 $, depending on the length of the segment $ OM $.