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Locus of a point

Source: APMO 1997

March 17, 2006
geometry unsolvedgeometry

Problem Statement

Triangle A1A2A3A_1 A_2 A_3 has a right angle at A3A_3. A sequence of points is now defined by the following iterative process, where nn is a positive integer. From AnA_n (n≥3n \geq 3), a perpendicular line is drawn to meet An−2An−1A_{n-2}A_{n-1} at An+1A_{n+1}. (a) Prove that if this process is continued indefinitely, then one and only one point PP is interior to every triangle An−2An−1AnA_{n-2} A_{n-1} A_{n}, n≥3n \geq 3. (b) Let A1A_1 and A3A_3 be fixed points. By considering all possible locations of A2A_2 on the plane, find the locus of PP.
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