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) 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.
2) 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.
Reference: 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!