The tie theorems
Abstract
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.
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