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34th Austrian Mathematical Olympiad 2003

Source: round 1, problem 3

February 13, 2009
geometry unsolvedgeometry

Problem Statement

Given are two parallel lines g g and h h and a point P P, that lies outside of the corridor bounded by g g and h h. Construct three lines g1 g_1, g2 g_2 and g3 g_3 through the point P P. These lines intersect g g in A1,A2,A3 A_1,A_2, A_3 and h h in B1,B2,B3 B_1, B_2, B_3 respectively. Let C1 C_1 be the intersection of the lines A1B2 A_1B_2 and A2B1 A_2B_1, C2 C_2 be the intersection of the lines A1B3 A_1B_3 and A3B1 A_3B_1 and let C3 C_3 be the intersection of the lines A2B3 A_2B_3 and A3B2 A_3B_2. Show that there exists exactly one line n n, that contains the points C1,C2,C3 C_1,C_2,C_3 and that n n is parallel to g g and h h.
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