## 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 2*n*-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.