engleski

On improvements of the Butterfly theorem

This paper explores the locus of butterfly points of a quadrangle ABCD in the plane. These are the common midpoints of three segments formed from intersections of a butterfly line with the lines AB, CD, AD, BC, AC, and BD. The locus is the nine-point-conic of ABCD that goes through the midpoints of the segments AB, CD, AD, BC, AC, and BD. We also consider the problem to determine when two quadrangles share the nine-point-conic. Our proofs use analytic geometry of the rectangular Cartesian coordinates.

quadrangle; butterfly theorem; conic; locus; nine-point-conic; cyclic quadrangle; Kiepert; Jarabek; Feuerbach; center; central point

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