2021/02/22 by Horwitz, Alan
#97G70 #97G99 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2102.11369
A convex quadrilateral, Q, is called a midpoint diagonal quadrilateral if the intersection point of the diagonals of Q coincides with the midpoint of at least one of the diagonals of Q. A parallelogram, P, is a special case of a midpoint diagonal quadrilateral since the diagonals of P bisect one another. We prove two results about ellipses inscribed in midpoint diagonal quadrilaterals, which generalize properties of ellipses inscribed in parallelograms involving convex quadrilaterals. First, Q is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in Q has tangency chords which are parallel to one of the diagonals of Q. Second, Q is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in Q has a unique pair of conjugate diameters parallel to the diagonals of Q. Finally, we show that there is a unique ellipse, EI, of minimal eccentricity inscribed in a midpoint diagonal quadrilateral, Q, and also that the unique pair of conjugate diameters parallel to the diagonals of Q are the equal conjugate diameters of EI.