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.
Note: The above result 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’.
Reference: Coxeter, H.S.M, Greitzer, S.L. (1967). Pedal Triangles - pp. 23-25. Geometry Revisited. Washington, DC: The Mathematical Association of America.

(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-out' 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?

Published Paper
A paper Optimising Miquel circles centre triangles and third pedal triangles by Hans Humenberger (University of Vienna) & myself, with proofs of these results, has been published Online Open Access, 19 Jan 2026, in The Mathematical Gazette. DOI: 10.1080/00255572.2025.2539599.

Related Links
A generalization of Neuberg's Pedal Theorem and the Simson-Wallace line
Miquel's Theorem (Rethinking Proof activity)
A variation of Miquel and its generalization
Minimum Area of Miquel Circles Centres Triangle
Varignon Parallelogram Area (Rethinking Proof activity)
Area ratios of some parallel-polygons inscribed in quadrilaterals and triangles
(above webpage explores the areas of the general Varignon parallelogram & Thomsen's hexagon)
Maximizing the Area of Van Aubel's Quadrilateral
Maximizing the Area of Napoleon's Triangle
Equi-inclined Lines Problem
Generalizations of a theorem by Wares
Carnot's (or Bottema's) Perpendicularity & Areas Theorem & Some Generalizations
Power Lines of a Triangle
Power Lines Special Case: Altitudes of a Triangle
Distances in an Equilateral Triangle (Viviani's theorem) (Rethinking Proof activity)
Sums of alternate triangular areas in regular & semi-regular side-2n-gons (Application of Viviani)
Clough's Theorem (a variation of Viviani) and some Generalizations
2D Generalizations of Viviani's Theorem
Parallelogram Distances

External Links
Joseph Jean Baptiste Neuberg (Biography on Wikipedia)
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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