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Pascal-points quadrilaterals inscribed in a cyclic quadrilateral

Published online by Cambridge University Press:  06 June 2019

David Fraivert
David Fraivert*
Affiliation:
Department of Mathematics, Shaanan College, P.O. Box 906, Haifa 26109, Israel e-mail: [email protected]
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Extract

This paper presents some new theorems about the Pascal points of a quadrilateral. We shall begin by explaining what these are.

Let ABCD be a convex quadrilateral, with AC and BD intersecting at E and DA and CB intersecting at F. Let ω be a circle through E and F which meets CB internally at M and DA internally at N. Let CA meet ω again at L and let DB meet ω again at K. By using Pascal’s theorem for the crossed hexagons EKNFML and EKMFNL and which are circumscribed by ω, the following results can be proved [1]:–

  1. (a) NK, ML and AB are concurrent (at a point P internal to AB)

  2. (b) NL, KM and CD are concurrent (at a point Q internal to CD)

Type
Articles
Information
The Mathematical Gazette , Volume 103 , Issue 557 , July 2019 , pp. 233 - 239
Copyright
© Mathematical Association 2019 

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References

Fraivert, D., The theory of a convex quadrilateral and a circle that forms Pascal points − the properties of Pascal points on the sides of a convex quadrilateral, Journal of Mathematical Sciences: Advances and Applications 40 (2016) pp. 1-34, accessed January 2019 at: http://dx.doi.org/10.18642/jmsaa_7100121666Google Scholar
Fraivert, D., Properties of a Pascal points circle in a quadrilateral with perpendicular diagonals, Forum Geometricorum, 17 (2017) pp. 509-526. http://forumgeom.fau.edu/FG2017volume17/FG201748.pdfGoogle Scholar