A generalization of Neuberg's Theorem and the Simson line

A generalization of Neuberg's 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) 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.
References
Coxeter, H.S.M, Greitzer, S.L. (1967). 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.

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?
3) Can you generalize further to quadrilaterals? Explore!