## 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.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } 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!

Explore the further or related generalizations below and also try to explain why (prove) they are true.

A generalization of Neuberg's Theorem to polygons

Related Generalizations of Viviani's Theorem

A generalization of the Simson-Wallace line (1799)
From point P on the circumcircle of ΔABC construct lines to the sides (or their extensions) of ΔABC so that they respectively form three equal angles with the sides. Then these three "feet" lie on a straight line, the 'Miquel line'.

Note
Read more about the historical background of the Simson-Wallace line, named after two 18th century British mathematicians.

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

Explore the further or related generalizations below and also try to explain why (prove) why they are true.

More properties of the generalised Simson-Wallace line

Miquel Deltoid (or Hypocycloid)

Ellipse by Reference Triangle (Kindly sent to me in January 2010 by Hungarian geophysicist Sandor Szanto, e-mail: szanto.danielne@upcmail.hu)

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

Related Investigation
The paper below explores some properties of several sequences of nested triangles, among which is Neuberg's theorem.
Ismailescu, D. & Jacobs, J. (2006). On Sequences of Nested Triangles. Periodica Mathematica Hungarica, Vol. 53 (1–2), pp. 169–184.

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.