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AC, BD and \lambd concurrent or //, 4 circles related

Source: 2018 NZOMC Camp Selections p8

January 10, 2021
geometryconcurrentparallel

Problem Statement

Let λ\lambda be a line and let M,NM, N be two points on λ\lambda. Circles α\alpha and β\beta centred at AA and BB respectively are both tangent to λ\lambda at MM, with AA and BB being on opposite sides of λ\lambda. Circles γ\gamma and δ\delta centred at CC and DD respectively are both tangent to λ\lambda at NN, with CC and DD being on opposite sides of λ\lambda. Moreover AA and CC are on the same side of λ\lambda. Prove that if there exists a circle tangent to all circles α,β,γ,δ\alpha, \beta, \gamma, \delta containing all of them in its interior, then the lines AC,BDAC, BD and λ\lambda are either concurrent or parallel.
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