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.
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
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!
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
Ellipse by Reference Triangle (Kindly sent to me in January 2010 by Hungarian geophysicist Sandor Szanto, e-mail: email@example.com)
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.
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.