Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem

Neuberg's Theorem (1892): From point P construct perpendicular lines to the sides (or their extensions) of ΔA₀B₀C₀. Repeat the same process from P to the Miquel ΔA₁B₁C₁, and again to Miquel ΔA₂B₂C₂. Then ΔA₃B₃C₃ is similar to ΔA₀B₀C₀.
(Below is a dynamic geometry sketch illustrating the theorem - also see A generalization of Neuberg's Theorem and the Simson line, using equi-inclined lines to the sides).

Investigate
1) Use the sketch below and drag P to investigate where the third pedal ΔA₃B₃C₃ has maximum area.
2) Ensure that you drag P to coincide with I, the incentre of ΔA₀B₀C₀. What do you notice?
3) Drag any of A₀, B₀, or C₀. Then repeat steps 1) and 2).
4) Formulate a conjecture about the location of P in the interior of the initial triangle so that ΔA₃B₃C₃ has maximum area (with respect to a fixed ΔA₀B₀C₀).

Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem

Challenge
Can you explain why (prove that) that your conjecture in 4) above is true?

Explore More
What happens if P is dragged outside ΔA₀B₀C₀?
5) Click on the 'Link to Neuberg triangle with excentres' button. This sketch is a 'zoomed-in' version of the preceding one showing the excentres E₁, E₂ and E₃.
6) Drag P to each of the excentres E₁, E₂ and E₃. What do you notice?
7) For a fixed ΔA₀B₀C₀, write down the area of ΔA₃B₃C₃ at each of I, E₁, E₂ and E₃. Can you find a relationship between all four measurements?
8) Challenge: Can you explain why (prove that) your observations in 6) and 7) are true?

Submitted Paper
A paper "Optimizing Triangle Areas: Miquel Circles Center Triangle and 3rd Pedal Triangle" by Hans Humenberger & myself, with proofs of these results, has been submitted for publication. All Rights Reserved.

Related Links
A variation of Miquel and its generalization
A generalization of Neuberg's Theorem and the Simson line
Minimum Area of Miquel Circles Centres Triangle
Equi-inclined Lines Problem
Generalizations of a theorem by Wares