OIM 1992 Problem 3

Let $ABC$ be an equilateral triangle of sidelength 2 and let $\omega$ be its incircle.

a) Show that for every point $P$ on $\omega$ the sum of the squares of its distances to $A$ , $B$ , $C$ is 5.

b) Show that for every point $P$ on $\omega$ it is possible to construct a triangle of sidelengths $AP$ , $BP$ , $CP$ . Also, the area of such triangle is $\frac{\sqrt{3}}{4}$ .