Given a point P and a triangle Δ A B C, the Miquel triangle is the triangle Δ P_A P_B P_C connecting the side points P_A, P_B, and P_C of Δ A B C with respect to which M is the Miquel point. Let the points defining the Miquel circles be fractional distances k_a, k_b, and k_c along the sides B C, C A, and A B, respectively, and let k_i^, = 1 - k_i and k_(i j)^, = 1 - k_i - k_j.
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