Sunday, 4 July 2010

A Pythagorean Generalization

Looking through the recent, and excellent, 67th Carnival of MathematicsI came across a link to a theorem at "Cut the Knot" that I had never known.

It begins with the simple fact that for a triangle ABC, Cos2(A) + Cos2(B) + Cos2(C) = 1 IFF (if and only if) the triangle is a right triangle..that is, one of Cos(A), Cos(B) or Cos(C) = 0.
But the part I loved is the more general extension... that for ANY triangle, Cos2(A) + Cos2(B) + Cos2(C)+2 Cos(A)Cos(B)Cos(C) = 1.

Since only one of the angles can be obtuse (and hence the quantity 2 Cos(A)Cos(B)Cos(C) would be negative only in the obtuse case), we can use Cos2(A) + Cos2(B) + Cos2(C) as a determinant for triangles. When the sum is > 1 the triangle is obtuse. If it is equal to one, the triangle is a right triangle; and if the sum is< 1, the triangle is acute. Not sure how I got so old without knowing that.

Can anyone tell me who/when this general identity was first discovered?
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