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MMO 252 Moscow MO 1953 concurrency, angles \pi - (given to a line)

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August 9, 2019
anglesconcurrentLinegeometry

Problem Statement

Given triangle △A1A2A3\vartriangle A_1A_2A_3 and a straight line ℓ\ell outside it. The angles between the lines A1A2A_1A_2 and A2A3,A1A2A_2A_3, A_1A_2 and A2A3,A2A3A_2A_3, A_2A_3 and A3A1A_3A_1 are equal to a3,a1a_3, a_1 and a2a_2, respectively. The straight lines are drawn through points A1,A2,A3A_1, A_2, A_3 forming with ℓ\ell angles of π−a1,π−a2,π−a3\pi -a_1, \pi -a_2, \pi -a_3, respectively. All angles are counted in the same direction from ℓ\ell . Prove that these new lines meet at one point.
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