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Area of triangles, eventually represented by complex numbers

Source: Romanian National Olympiad 2016, grade x, p. 3

August 25, 2019
Complex Geometryanalytic geometrygeometryPure geometrycomplex numbers

Problem Statement

a) Let be two nonzero complex numbers a,b. a,b. Show that the area of the triangle formed by the representations of the affixes 0,a,b 0,a,b in the complex plane is 14∣a‾b−ab‾∣. \frac{1}{4}\left| \overline{a} b-a\overline{b} \right| .
b) Let be an equilateral triangle ABC, ABC, its circumcircle C, \mathcal{C} , its circumcenter O, O, and two distinct points P1,P2 P_1,P_2 in the interior of C. \mathcal{C} . Prove that we can form two triangles with sides P1A,P1B,P1C, P_1A,P_1B,P_1C, respectively, P2A,P2B,P2C, P_2A,P_2B,P_2C, whose areas are equal if and only if OP1=OP2. OP_1=OP_2.
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