MathDB

47

Part of 1983 IMO Longlists

Problems(1)

If angles are equal to pi/3 then lines are concurent

Source:

10/7/2010
In a plane, three pairwise intersecting circles C1,C2,C3C_1, C_2, C_3 with centers M1,M2,M3M_1,M_2,M_3 are given. For i=1,2,3i = 1, 2, 3, let AiA_i be one of the points of intersection of CjC_j and Ck ({i,j,k}={1,2,3})C_k \ (\{i, j, k \} = \{1, 2, 3 \}). Prove that if ∠M3A1M2=∠M1A2M3=∠M2A3M1=π3 \angle M_3A_1M_2 = \angle M_1A_2M_3 = \angle M_2A_3M_1 = \frac{\pi}{3}(directed angles), then M1A1,M2A2M_1A_1, M_2A_2, and M3A3M_3A_3 are concurrent.
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