Solution to Problem 28

Correct solutions came from Alejandro Vellano, Spain; Al Zimmermann, USA; Philippe Fondanaiche, France; Tegid Bard; Burkart Venzke, Germany; Jens Voβ, Germany; Gary Smith, USA; David Stignant, USA; Ahron Teitelman, Israel; Dan Dima, Romania; Ron Welch, USA; Lou Cairoli, USA; Jerome Cherry, Canada; Stefan Gatachiu, Romania; Farid Lian, Colombia; John Snyder, USA; Bill Webb, USA.

The value of the expression is always -3.

After an appropriate translation of the triangle, we may assume that one vertex is at the origin and the other two vertices lie above the x-axis.  Let a be the angle shown on the left with sides having slopes m1, m2, m3 as indicated.  It's straightforward to see that  m1 = tan(a), m2 = tan(a + p/3), m3 = tan(a + 2p/3).  The tangent angle-sum formula gives 

which after multiplying, adding, and simplifying yields m1m2 + m2m3 + m1m3 = -3.

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