A generalization of Neuberg's Pedal Theorem to polygons

A Generalization of Stewart's Theorem (1940)
From point P construct equi-inclined lines to the sides (or their extensions) of quadrilateral ABCD to form a Miquel quadrilateral. Repeat the same process from P to the Miquel quadrilateral, and three times more. Then the Miquel quadrilateral A₄B₄C₄D₄ is similar to quadrilateral ABCD.
Note: The proof of this equi-inclined lines generalization follows directly from the spiral similarity centred at P.

 

A generalization of Neuberg's Pedal Theorem to polygons

References
De Villiers, M. (2002). From nested Miquel triangles to Miquel distances. Math Gazette, 86(507), pp. 390-395.
Humenberger, H. & De Villiers, M. (2026). Optimising Miquel circles centre triangles and third pedal triangles. The Mathematical Gazette. DOI: 10.1080/00255572.2025.2539599
Stewart, B.M. (1940). Cyclic Properties of Miquel Polygons. Am. Math. Monthly, Vol. 47 (Aug-Sept.), pp. 462-466.

Other examples involving equi-inclined lines
A generalization of Neuberg's Pedal Theorem & the Simson-Wallace line
Further generalizations of Viviani's Theorem
Equi-inclined Lines Problem
Generalizations of a theorem by Wares
Equi-inclined Lines to the Sides of a Quadrilateral at its Vertices
The Equi-inclined Bisectors of a Cyclic Quadrilateral
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral

Related Links
Miquel's Theorem (Rethinking Proof activity)
A variation of Miquel's theorem and its generalization
Minimum Area of Miquel Circle Centres Triangle
A generalization of Neuberg's Pedal Theorem to Polygons
Maximising the Area of the 3rd Pedal Triangle in Neuberg's pedal theorem
Distances in an Equilateral Triangle (Viviani's theorem) (Rethinking Proof activity)
2D Generalizations of Viviani's Theorem (Equilateral or equi-angled polygons or polygons with opposite sides parallel)
Further generalizations of Viviani's Theorem (Using equi-inclined lines)
Clough's Theorem (a variation of Viviani) and some Generalizations

External Links
Pedal triangle (Wikipedia)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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