RO-NMO 2016 Problem 103

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