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On the Locus of a Pedal Point of a Quadrilateral Such That Opposite Sides of the Pedal Quadrilateral Form a Constant Angle

Author

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  • Jiří Blažek
  • Pavel Pech

Abstract

The article deals with a problem related to plane quadrilaterals, which is connected with the well-known Wallace–Simson theorem. If K,L,M,N are the feet of perpendiculars from a point P to the sides of a quadrilateral, then the locus of P such that lines KN and LM form a constant angle is generally a circle. It turns out that the proof based on finding the equation of the locus of P is quite complex, time-consuming, and requires human intervention in addition to a computer. Therefore, we decided to present a classical geometric proof using the Miquel point of a quadrilateral. Furthermore, the properties of a cyclic quadrilateral are studied in a similar way to the properties of a triangle in the Wallace–Simson theorem.

Suggested Citation

  • Jiří Blažek & Pavel Pech, 2026. "On the Locus of a Pedal Point of a Quadrilateral Such That Opposite Sides of the Pedal Quadrilateral Form a Constant Angle," Journal of Mathematics, Hindawi, vol. 2026, pages 1-6, March.
    • Handle: RePEc:hin:jjmath:5239001
    DOI: 10.1155/jom/5239001
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