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concurrent lines by a triangle ,a circle and 3 products

Source: Czech-Polish-Slovak Match 2014 day 2 P1

October 2, 2017
geometryconcurrent

Problem Statement

Let ABCABC be a triangle, and let PP be the midpoint of ACAC. A circle intersects AP,CP,BC,ABAP, CP, BC, AB sequentially at their inner points K,L,M,NK, L, M, N. Let SS be the midpoint of KLKL. Let also 2⋅∣AN∣⋅∣AB∣⋅∣CL∣=2⋅∣CM∣⋅∣BC∣⋅∣AK∣=∣AC∣⋅∣AK∣⋅∣CL∣.2 \cdot | AN |\cdot |AB |\cdot |CL | = 2 \cdot | CM | \cdot| BC | \cdot| AK| = | AC | \cdot| AK |\cdot |CL |. Prove that if P≠SP\ne S, then the intersection of KNKN and MLML lies on the perpendicular bisector of the PSPS.
(Jan Mazák)
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