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MMO 334 Moscow MO 1956 4 perpendiculars in regular hexagon

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August 19, 2019
geometryperpendicularhexagon

Problem Statement

a) Points A1,A2,A3,A4,A5,A6A_1, A_2, A_3, A_4, A_5, A_6 divide a circle of radius 11 into six equal arcs. Ray ℓ1\ell_1 from A1A_1 connects A1A_1 with A2A_2, ray ℓ2\ell_2 from A2A_2 connects A2A_2 with A3A_3, and so on, ray ℓ6\ell_6 from A6A_6 connects A6A_6 with A1A_1. From a point B1B_1 on ℓ1\ell_1 the perpendicular is drawn on ℓ6\ell_6, from the foot of this perpendicular another perpendicular is drawn on ℓ5\ell_5, and so on. Let the foot of the 66-th perpendicular coincide with B1B_1. Find the length of segment A1B1A_1B_1.
b) Find points B1,B2,...,BnB_1, B_2,... , B_n on the extensions of sides A1A2,A2A3,...,AnA1A_1A_2, A_2A_3,... , A_nA_1 of a regular nn-gon A1A2...AnA_1A_2...A_n such that B1B2⊥A1A2B_1B_2 \perp A_1A_2, B2B3⊥A2A3B_2B_3 \perp A_2A_3,... . . . , BnB1⊥AnA1B_nB_1 \perp A_nA_1.
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