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 ΔA_{0}B_{0}C_{0}. Repeat the same process from P to the Miquel ΔA_{1}B_{1}C_{1}, and again to Miquel ΔA_{2}B_{2}C_{2}. Then ΔA_{3}B_{3}C_{3} is similar to ΔA_{0}B_{0}C_{0}.
(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 ΔA_{3}B_{3}C_{3} has maximum area.
2) Ensure that you drag P to coincide with I, the incentre of ΔA_{0}B_{0}C_{0}. What do you notice?
3) Drag any of A_{0}, B_{0}, or C_{0}. Then repeat steps 1) and 2).
4) Formulate a conjecture about the location of P in the interior of the initial triangle so that ΔA_{3}B_{3}C_{3} has maximum area (with respect to a fixed ΔA_{0}B_{0}C_{0}).

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 ΔA_{0}B_{0}C_{0}?
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 E_{1}, E_{2} and E_{3}.
6) Drag P to each of the excentres E_{1}, E_{2} and E_{3}. What do you notice?
7) For a fixed ΔA_{0}B_{0}C_{0}, write down the area of ΔA_{3}B_{3}C_{3} at each of I, E_{1}, E_{2} and E_{3}. 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