Steiner Lehrsatz 4, bd I, p. 177
Apply three times the theorem of Pappus and get three lines (GHI), (KLM) and (NOP). They pass all through the same point.
(GHI) is invariant under the homography, constructible through the tool [ Transforms \ Homography 1 conic _ ], which maps A, B, C to F, E, D correspondingly. Analogous statements hold for the other lines (KLM) and (NOP) too.
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