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!
A generalization of Neuberg's Theorem to polygons
Related Generalizations of Viviani's Theorem
Important: To view & manipulate the DYNAMIC version of the Simson-Wallace generalization, navigate to it using the appropriate button in the dynamic sketch right at the TOP; the picture below is static.
A generalization of the Simson-Wallace line
More properties of the generalised Simson-Wallace line
Miquel Deltoid (or Hypocycloid)
Published Paper
Download an article of mine from the Mathematical Gazette (2002) giving proofs of the results above at From nested Miquel triangles to Miquel distances.
Explore More
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem
More Equi-inclined lines
Other examples of results involving equi-inclined lines:
A variation of Miquel and its generalization
Equi-inclined Lines Problem
Generalizations of a theorem by Wares
Other Related Links
Carnot's (or Bottema's) Perpendicularity Theorem & Some Generalizations
Power Lines of a Triangle
Power Lines Special Case: Altitudes of a Triangle
Clough's Theorem (a variation of Viviani) and some Generalizations
2D Generalizations of Viviani's Theorem
Parallelogram Distances
Some Variations of Vecten configurations
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem
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Created by Michael de Villiers, 21 January 2010. Changed to WebSketchpad, 21 April 2020; modified 9 August 2021; updated 14 May 2023; 20 July 2023; 18 March 2025.