## Clough's Theorem (a variation of Viviani) and some Generalizations

During 2003, Duncan Clough, a Grade 11 student from Bishops Diocesan College, a high school in Cape Town, was exploring Viviani's Theorem, which says that the sum of distances of a point to the sides of an equilateral triangle is constant. Using dynamic geometry software, he then discovered (but could not himself prove) the following interesting variation of Viviani's theorem.

Clough's Theorem: Label the feet of the altitudes from an arbitrary point P inside an equilateral triangle ABC to the sides AB, BC, AC respectively as PC, PA, PB, then APC + BPA + CPB is constant. (Drag point P in the sketch, or B or C to change the size or orientation of the triangle.)

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Clough's Theorem (a variation of Viviani)

Challenge
Can you logically explain (prove) why the result is true? Can you find different explanations (proofs)?

Exploring Some Generalizations
Viviani's Theorem generalizes to polygons that are equilateral or equi-angled, or to 2n-gons with opposite sides parallel - see for example, 2D Generalizations of Viviani's Theorem.
a) Can you similarly generalize Clough's Theorem to these higher polygons? Click on the LINK buttons in the sketch above to check.
b) Can you also logically explain (prove) why these generalizations are also true?

Some Proofs
1) Read my ICME 12 Regular Lecture in Korea in July 2012, published in Pythagoras, An illustration of the explanatory and discovery functions of proof for various proofs of the above.
2) Read the 2012 proof of Clough's Theorem by Shailesh Shirali from India in the mathematics education journal "At Right Angles" at: Viviani's Theorem and a Cousin.
3) Clough's Theorem was used as a problem in the first round of the British Mathematical Olympiad of of 2013/2014. Read one of the proofs produced by students, which was similar to the following very short and elegant one, kindly sent to me in 2015 by Gregoire Nicollier from Switzerland at: Clough's Theorem: The Simplest Proof.
4) The above proof, together with a dynamic sketch, also appears at the Cut The Knot site at: Clough's Theorem: The Simplest Proof.

Other Related Items & Papers
1) In the Oct 2021 Problem Corner of the Electronic Journal of Mathematics & Technology (eJMT), a generalization of Clough's theorem to a regular n-gon was posed as one of the two problems.
2) Here are two sets of solutions that were provided in eJMT for the two Problem Corner problems: a) synthetic proofs and b) alternative computer assisted proofs using GeoGebra Discovery.
The following recent conference paper also has an initial discussion of Clough's theorem:
3) Kovács, Z.; Recio, T.; Vélez, M.P. (2021). Approaching Cesàro’s inequality through GeoGebra Discovery. Proceedings of the 26th Asian Technology Conference in Mathematics, W.C. Yang, D.B. Meade, M. Majewski (eds). Published by Mathematics and Technology, LLC. ISSN 1940-4204, Dec. 13-15, 2021. pp. 160-174.

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Created by Michael de Villiers, 23 January 2013; updated 23 June 2015; 14 April 2019; 13 March 2023.