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Area of a triangle given some condition upon complex numbers

Source: Romanian National Olympiad 2003, grade x, p. 3

August 27, 2019
geometryComplex Geometrycomplex numbers

Problem Statement

Let be a circumcircle of radius 1 1 of a triangle whose centered representation in the complex plane is given by the affixes of a,b,c, a,b,c, and for which the equation a+bcos⁡x+csin⁡x=0 a+b\cos x +c\sin x=0 has a real root in (0,π2). \left( 0,\frac{\pi }{2} \right) . prove that the area of the triangle is a real number from the interval (1,1+22]. \left( 1,\frac{1+\sqrt 2}{2} \right] .
Gheorghe Iurea
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