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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

480.0. "What is the theorem about the inscribed hexagon?" by KEEPER::KOSTAS (Kostas G. Gavrielidis <o.o> ) Thu May 01 1986 02:06

    Hello,
     
         The question is this:
    
         "What is the theorem about the inscribed hexagon?"
    
    
    Enjoy,
    
    Kostas G.
T.RTitleUserPersonal
Name		DateLines
480.1glad you asked, although I don't know whyMETOO::YARBROUGHThu May 01 1986 12:5910
    "A hexagon that is not inscribed in anything is free to be any shape
    it wants to." (Yarbrough's relaxed hexagon theorem.)
    
    "The regular hexagon inscribed in the triangle (1,2,3) is the smallest of
    all hexagons." (Yarbrough's litle theorem.)
                 
    "The hexagon with null sides can be inscribed in anything."
    (Yarbrough's general hexagon theorem.)
                            
    Take your pick.
480.2Blaise Pascal (1623-1662) KEEPER::KOSTASKostas G. Gavrielidis <o.o> Mon May 05 1986 12:4116
    Well,
    
        Blaise Pascal (1623-1662) at the age of 15 discovered that the
        opposite sides of an inscribed hexagon intersect in three collinear
        points. He also found 400 corollaries.
    
    Now the theorem is very general.
    
        As hexagon we may take any six points, a b' c a' b c',  on any
        conic (ellipse, parabola, or hyperbola) connect them in this
        order - whatever their order on the conic- and take as opposite
        sides the pairs:
    
             (ab',a'b), (bc',b'c), (ca',c'a).