Thứ Bảy, 19 tháng 1, 2019

IMO 1963 - Problem 2

Problem: Find the locus of points in space that are vertices of right angles of which one ray passes through a given point and the other intersects a given segment.
Solution: Let $A$ be the given point, $BC$ the given segment, and $\mathscr{B_1},\mathscr{B_2}$ the closed balls with the diameters $AB$ and $AC$ respectively. Consider one right angle $\angle{AOK}$ with $K\in [BC]$. If $B',C'$ are the feet of the perpendiculars from $B,C$ to $AO$ respectively, then $O$ lies on the segment $B'C'$, which implies that it lies on exactly one of the segment $AB',AC'$. Hence $O$ belongs to exactly one of the balls $\mathscr{B_1},\mathscr{B_2},i.e, O\in \mathscr{B_1}\Delta \mathscr{B_2}$. This is obviously the required locus.
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