**A generalization of Neuberg's Theorem (1892)**

From point *P* construct lines to the sides (or their extensions) of Δ*ABC* so that they form *equal angles* with the sides (e.g. ∠*PA _{1}B* = ∠

**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.

A generalization of Neuberg's Theorem

**Challenge**

1) Drag any of *A*, *B*, *C*, *P*, *C _{1}*,

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

From point

**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

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.

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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.