Given $A(3+4i)$, $B(-4+3i)$ and $C(4+3i)$ be the vertices of a triangle $ABC$ which is inscribed in a circle $S=0$. Let $AD, BE, CF$ be altitudes through $A, B, C$ which meet the circle S=0 at $$D(z_1), E(z_2)\,and\,F(z_3)$$ respectively, then find the value of $${z_1}{z_2}{z_3}$$
I have actually solved this question by finding the co-ordinates of points $D, E$ and $F$ and simply multiplying the complex numbers. But this method was quite lengthy and it was difficult to find the co-ordinates of point $F$. The answer comes out to be quite simple $(75 + 100i$). Do any simpler and more elegant methods exist?