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.)
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 to ...' buttons in the sketch above to check.
b) Can you also logically explain (prove) why these generalizations in a) are also true?
c) Click on the 'Link to Alternate Areas Sum' button to navigate to a new sketch showing an 'alternate area variation' related to Clough's theorem.
d) What do you notice in c)? Can you explain why (prove) this is true? Can you generalize further?
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 Clough's theorem and its generalizations to higher polygons.
2) Work through a NCTM & AMESA Workshop Guide, Clough's Conjecture: A Sketchpad Investigation, which was used in 2h workshops conducted with mathematics teachers at NCTM in April 2014 and at AMESA in July 2014. An accompanying Sketchpad sketch to be used with the guide can be downloaded as a zipped file from Clough Workshop GSP zipped file. (Note that the sketch can be opened (after unzipping) with the free Sketchpad 5 version available at the link at the bottom of this page).
3) Read the Dec 2012 proofs of Clough's Theorem by Shailesh Shirali from India in the mathematics education journal "At Right Angles" at: Viviani's Theorem and a Cousin.
4) Clough's Theorem was used as a problem in the first round of the British Mathematical Olympiad 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.
5) 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) J.N. Contreras (2014) in Discovering and Extending Viviani’s Theorem with GeoGebra not only discusses Viviani's theorem, but Clough's theorem as well as the 'alternate areas variation' mentioned in my ICME 12 lecture & illustrated in the dynamic sketch above.
2) 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.
3) 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:
4) 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.
Related Links
Distances in an Equilateral Triangle (Viviani's theorem) (Rethinking Proof activity)
2D Generalizations of Viviani's Theorem
Further generalizations of Viviani's Theorem (Using equi-inclined lines)
Parallelogram Distances
3D Generalizations of Viviani's Theorem
Carnot's (or Bottema's) Perpendicularity & Areas Theorem & Some Generalizations
Vreken's sum of triangular areas in a regular hexagon (Application of Viviani)
Sums of alternate triangular areas in regular & semi-regular side-2n-gons (Application of Viviani)
Triangle Altitudes (Rethinking Proof activity)
Miquel's Theorem (Rethinking Proof activity)
A variation of Miquel's theorem and its generalization (uses equi-inclined lines)
Minimum Area of Miquel Circle Centres Triangle
A generalization of Neuberg's Pedal Theorem & the Simson line (using equi-inclined lines)
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem
Some Trapezoid Explorations (See Investigation 5: Perpendicular Bisectors of Trapezoid)
The Perpendicular Bisectors of a Parallelogram
The Perpendicular Bisectors of an Apollonius Quadrilateral
Light Ray in a Triangle (Fagnano's Minimal Path) (Rethinking Proof activity)
Some External Links
Clough's Theorem: The Simplest Proof (Cut The Knot)
Viviani's theorem (Wikipedia)
Vincenzo Viviani (Wikipedia)
Vincenzo Viviani (MacTutor)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
********************************
Back to "Dynamic Geometry Sketches"
Back to "Student Explorations"
Created by Michael de Villiers, 23 January 2013 with JavaSketchpad; converted to WebSketchpad, 23 June 2015; updated 14 April 2019; 13 March 2023; 20 Nov 2025; 13 July 2026; 5 Sept 2026.