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sequence of triangles of from angle trisectors. // sides wanted in triangles

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 Shortlist G3 day1

September 29, 2021
geometryparalleltrisectortrisect

Problem Statement

Let △A1B1C1\vartriangle A_1B_1C_1 and l1,m1,n1l_1, m_1, n_1 be the trisectors closest to A1B1A_1B_1, B1C1B_1C_1, C1A1C_1A_1 of the angles A1,B1,C1A_1, B_1, C_1 respectively. Let A2=l1∩n1A_2 = l_1 \cap n_1, B2=m1∩l1B_2 = m_1 \cap l_1, C2=n1∩m1C_2 = n_1 \cap m_1. So on we create triangles △AnBnCn\vartriangle A_nB_nC_n . If △A1B1C1\vartriangle A_1B_1C_1 is equilateral prove that exists n∈Nn \in N, such that all the sides of △AnBnCn\vartriangle A_nB_nC_n are parallel to the sides of △A1B1C1\vartriangle A_1B_1C_1.
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