Question 9
Given an acute-angled triangle $ABC$ with area $S$, define $A^{\prime}$ as the intersection of the altitude line from $A$ to $BC$ with the outward-facing semicircle on diameter $BC$. Define $B^{\prime},C^{\prime}$ cyclically. Evaluate $T=[BCA^{\prime}]^2+[CAB^{\prime}]^2+[ABC^{\prime}]^2$, where brackets denote triangle area.