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Hail to the King - coplanar lines given and wanted, hexagonal pyramid, sphere

Source: 2022 Sharygin CR p24

March 6, 2022
geometry3D geometrypyramidsphere

Problem Statement

Let OABCDEFOABCDEF be a hexagonal pyramid with base ABCDEFABCDEF circumscribed around a sphere ω\omega. The plane passing through the touching points of ω\omega with faces OFAOFA, OABOAB and ABCDEFABCDEF meets OAOA at point A1A_1, points B1B_1, C1C_1, D1D_1, E1E_1 and F1F_1 are defined similarly. Let ℓ\ell, mm and nn be the lines A1D1A_1D_1, B1E1B_1E_1 and C1F1C_1F_1 respectively. It is known that ℓ\ell and mm are coplanar, also mm and nn are coplanar. Prove that ℓ\ell and nn are coplanar.
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