Question 59
a) Let $O$ be the origin of the complex plane and consider the points $A$ and $B$, whose complex coordinates are $a$ and $b$, respectively. Prove that $[OAB] = \frac{1}{4} |\bar{a}b - a\bar{b}|$, where $[OAB]$ denotes the area of triangle $OAB$.
b) Let $ABC$ be an equilateral triangle, $C$ its circumcircle, and $O$ its circumcenter. If a point $P$ lies in the interior of $C$, let $S(P)$ denote the area of the triangle whose side lengths equal the distances from $P$ to the triangle's vertices². Let $P_1$ and $P_2$ be two distinct points in the interior of $C$. Prove that $S(P_1) = S(P_2)$ if and only if $OP_1 = OP_2$.
²The existence of such a triangle is a famous result of the Romanian mathematician **Dimitrie Pompeiu** (1873-1954).
b) Let $ABC$ be an equilateral triangle, $C$ its circumcircle, and $O$ its circumcenter. If a point $P$ lies in the interior of $C$, let $S(P)$ denote the area of the triangle whose side lengths equal the distances from $P$ to the triangle's vertices². Let $P_1$ and $P_2$ be two distinct points in the interior of $C$. Prove that $S(P_1) = S(P_2)$ if and only if $OP_1 = OP_2$.
²The existence of such a triangle is a famous result of the Romanian mathematician **Dimitrie Pompeiu** (1873-1954).