User:DIT cook/sandbox

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"Desargues is not just a theorem, it's a lifestyle" - people who want to sound really cool and take ownership or something.

Desargues Involution Theorem (DIT): All conics passing through 4 fixed points intersect a fixed line in pairs under some involution depending on the 4 points and the line.

Dual Desargues Involution Theore (DDIT): All conics tangent to 4 fixed lines has tangents to a fixed point that are pairs under some involution of lines through that point.

Proof: Exercise for the reader

trivial exaple: Let ABC be triangle and I incentert. INcircle is tangnet to BC at D and D'¨is the antipode in the incircle wrt. D. Thje line AD' cutrs BC at K. PRove that BC and DK are shbare same midpoint. Soluton: Use dual desargues ivolution theroem (DDIT) pn quaidralteal ABDC from D'. TGenb there is an inovlutin swapping (D'B,D'C),(D'A,D'D), (D'D',D'D'). Projecting thorugh D' onto the line BC, there is an iunvlkution swqaooubg (B,C),(D,K),(infinty,infity). Since infinty is swapped with infiyty, the invoollutiop must be a reflection. Hence BC and DK share m,idpoint, as wamnted. (Who let him cook?)