Olympiad Number Theory · Superchallenge problem

Points $a = 1$, $b = e^{2\pi i/3}$, $c = e^{4\pi i/3}$ lie on the unit circle. The foot of the perpendicular from $p = 0.5$ (a real number) to chord $bc$ is $\dfrac12(b + c + p - bc\bar p)$. Compute the real part of that foot. Express your answer as a common fraction.