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concyclic, ext.angle bisectors of quad - All-Russian MO 2000 Regional (R4) 10.7

Source:

September 26, 2024
geometryangle bisectorConcyclic

Problem Statement

In a convex quadrilateral ABCDABCD we draw the bisectors a\ell_a, b\ell_b, c\ell_c, d\ell_d of external angles AA, BB, CC, DD respectively. The intersection points of the lines a\ell_a and b\ell_b, b\ell_b and c\ell_c, c\ell_c and d\ell_d, d\ell_d and a\ell_a are designated by KK, LL, MM, NN. It is known that 33 perpendiculars drawn from KK on ABAB, from LL om BCBC, from MM on CDCD intersect at one point. Prove that the quadrilateral ABCDABCD is cyclic.