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locus of lines' intersection from Cono Sur, condition 1/TH= 1/d -1/PH

Source: Cono Sur Shortlist 2003 G6

July 26, 2019
geometryLocusParallel Lines

Problem Statement

Let L1L_1 and L2L_2 be two parallel lines and L3L_3 a line perpendicular to L1L_1 and L2L_2 at HH and PP, respectively. Points QQ and RR lie on L1L_1 such that QR=PRQR = PR (Q≠HQ \ne H). Let dd be the diameter of the circle inscribed in the triangle PQRPQR. Point TT lies L2L_2 in the same semiplane as QQ with respect to line L3L_3 such that 1TH=1d−1PH\frac{1}{TH}= \frac{1}{d}- \frac{1}{PH} . Let XX be the intersection point of PQPQ and THTH. Find the locus of the points XX as QQ varies on L1L_1.
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