1963 IMO Problems/Problem 2
Point and segment are given. Determine the locus of points in space which are the vertices of right angles with one side passing through , and the other side intersecting the segment .
Let be the circle with diameter , and let be the circle with diameter . Then the locus is simply the set of points inside either or , but not both.
To see this, suppose the right angle's ray that does not pass through intersects segment at . Then the right angle's vertex must lie on the circle with diameter . So, for a particular , the desired locus is a circle with diameter . Accounting for all possible , the total locus is the union of the circumferences of all circles that have a diameter , where is some point on .
As moves from to , the motion of the circle with diameter is continuous and fluid. Any point lying within but outside will eventually be intersected by this moving circle, since it went from inside to outside the circle. This applies similarly to all points inside but outside . Also, the points inside both and are never intersected by this moving circle, as it always stays inside.
(This proof sucks and needs some formalism)
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