Duality: magic in simple geometry #SoME2

Two inaccuracies: 2:33 explains the first property (2:16), not the second one (2:24) Narration at 5:52 should be “intersections of GREEN and orange lines“ Time stamps: 0:00 — Intro 0:47 — Polar transform 4:46 — Desargues's Theorem 6:29 — Pappus's Theorem 7:18 — Sylvester-Gallai Theorem 8:55 — Solving a hard problem 10:23 — Ending A note on the definition of the polar: though our definition doesn't work for the point at the circle center if we work on the usual plane, we can formally add a “line at infinity” and call it the polar of the circle center. In this model every two lines intersect in one point, if the lines were parallel this point will be at infinity, and all points at infinity lie on one “line at infinity”. This is actually one of the models for the real projective plane, and by assigning each point it’s polar with respect to a fixed circle we have established an one-to-one correspondence between points and lines in the projective plane. A more complete overview of duality in Olympiad geometry: A beautiful book that among many things extensively covers polar transform: “Geometric transformations II“ by I.M. Yaglom