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 P_C, P_A, P_B, then AP_C + BP_A + CP_B 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.

Created by Michael de Villiers, 24 January 2013.