MathNet 1992 — Part 01 Question 43

$ABC$ is an equilateral triangle with side $2$. Show that any point $P$ on the incircle satisfies $PA^{2} + PB^{2} + PC^{2} = 5$. Show also that the triangle with side lengths $PA$, $PB$, $PC$ has area $\sqrt{3}/4$.