The tie theorems

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Theorem. There are general position points A, B, C, P on the projective plane. Let Aₚ be the intersection point of lines AP and BC. Analogously define Bₚ and Cₚ. Take any points A₁, B₁, C₁ on AP, BP, CP, respectively. Let WC be the intersection point of A₁Bₚ and B₁Aₚ. Analogously define points Wₐ and WB. Then lines CWC, AWₐ and BWB pass through one point. We also generalize this theorem and find interesting related properties.