A generalization of Neuberg's Pedal Theorem and the Simson-Wallace line

A generalization of Neuberg's Pedal Theorem (1892)
From point P construct lines to the sides (or their extensions) of ΔABC so that they form equal angles with the sides (e.g. ∠PA₁B = ∠PB₁C = ∠PC₁A). Repeat the same process from P to the Miquel ΔA₁B₁C₁, and again to Miquel ΔA₂B₂C₂. Then ΔA₃B₃C₃ is similar to ΔABC.

Notes
1) The special case of the above result for perpendiculars dropped to the sides forming pedal triangles is attributed by Coxeter & Greitzer (1967, pp. 23-24) to J. Neuberg, who as the editor added it to the 6th edition (1892) of John Casey’s classic ‘A Sequel to the First Six Books of the Elements of Euclid’.
2) Read some historical background on Joseph Neuberg (1840-1926) after whom the above theorem is named for the case when the above lines are perpendiculars to the sides.
3) In my paper in the references below, I have called such equi-inclined lines that form equal angles with the sides, the Miquel lines, and the respective distances measured from the point P to the sides, the Miquel distances.

A generalization of Neuberg's Theorem

Challenge
1) Drag any of A, B, C, P, C₁, A₂, or B₃ to dynamically explore the result.
2) Can you explain (prove) your observations?
(Hint: Use the concept of spiral similarity.)
3) Can you generalize further to quadrilaterals, pentagons, etc.? Explore!

Further or Related generalizations
Explore the further or related generalizations below (and also explain why (prove) they are true).
A generalization of Neuberg's Pedal Theorem to Polygons
Related Generalizations of Viviani's Theorem

Important: To view & manipulate the DYNAMIC version of the Simson-Wallace generalization, navigate to it using the appropriate button in the dynamic sketch right at the top; the picture below is static.

A generalization of the Simson-Wallace line

Further or Related generalizations
Explore the further or related generalizations below (and also explain why (prove) why they are true).
More properties of the generalised Simson-Wallace line
Miquel Deltoid (or Hypocycloid)

Explore More
For a given or fixed ΔABC which position of point P in the interior maximises the area of the 3rd pedal triangle ΔA₃B₃C₃? Click on the link below to investigate this problem.
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem

References
Coxeter, H.S.M, Greitzer, S.L. (1967). Pedal Triangles - pp. 23-25. Geometry Revisited. Washington, DC: The Mathematical Association of America.
De Villiers, M. (2002). From nested Miquel triangles to Miquel distances. Math Gazette, 86(507), pp. 390-395.

Related Investigation
The paper below explores some properties of several sequences of nested triangles, among which is Neuberg's pedal theorem.
Ismailescu, D. & Jacobs, J. (2006). On Sequences of Nested Triangles. Periodica Mathematica Hungarica, Vol. 53 (1–2), pp. 169–184.

More Equi-inclined lines
Other examples of results involving equi-inclined lines:
A variation of Miquel and its generalization
Equi-inclined Lines Problem
Generalizations of a theorem by Wares
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 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
Triangle Altitudes (Rethinking Proof activity, concurrency)
Carnot's (or Bottema's) Perpendicularity Theorem & Some Generalizations
Power Lines of a Triangle
Power Lines Special Case: Altitudes of a Triangle
Parallelogram Distances
Some Variations of Vecten configurations

External Links
Pedal triangle (Wikipedia)
Miquel's theorem (Wikipedia)
Spiral similarity (Wikipedia)
Miquel's Point (Cut The Knot)
Spiral Similarity (Wolfram MathWorld)
Miquel's Theorem (Wolfram MathWorld)
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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