If this theorem holds for a semicircle, does it also hold for any sector of a circle?

Question

The theorem below holds for a semicircle (exactly half of a circle, AB is diameter). But does the same hold for any sector of a circle?

Answer 1

Say you move $J$ to $I$, then $J=I$ and $L=I$. Now we would have: $$ KI^2 +0 = HI^2$$ which does not hold (unles $K=H$).