Clough's Conjecture

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.

Conjecture: 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 Conjecture

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

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. Can you similarly generalize Clough's Conjecture to these higher polygons? Use this free dynamic geometry applet online at GEONexT to explore experimentally.

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Created by Michael de Villiers, 24 January 2013.