What question did this study set out to answer?

The aim is to identify the locus of pedal points in quadrilaterals where opposite sides form a constant angle.

March 16, 2026Open Access

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

Key Points

The aim is to identify the locus of pedal points in quadrilaterals where opposite sides form a constant angle.
Examined the properties of pedal points in the context of quadrilaterals.
Utilized the Miquel point in classical geometry for proof.
Compared findings with the Wallace–Simson theorem and cyclic quadrilateral properties.
Identified that the locus of pedal points generally forms a circle.
Established a classical geometric proof that simplifies the understanding of the problem.
Detailed the parallels between quadrilateral properties and those of triangles as per the Wallace–Simson theorem.

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 K N and L M 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.

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

Key Points

Abstract

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