An inequality involving a triangle and an interior point and its application

Let x0 be an interior split point in the triangle T := [x1,x2,x3] . By αi j we denote the angle ̂ x0,xi,x j , i = j . We show that cosα12 cosα23 cosα31 + cosα21 cosα32 cosα13 > 0. Additionally, we use this inequality to prove uniqueness and existence of a conforming quadratic piecewise harmonic finite element on the Clough-Tocher split of a triangle. Mathematics subject classification (2010): 51N20, 65N30.