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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?

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  • $\begingroup$ Observe that the center of the circle is at $0$ with radius $5$, and that multiplication "rotates" coordinates. $\endgroup$ – chubakueno May 13 '14 at 7:20
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    $\begingroup$ What do you mean by "the circle $S = 0$"? $\endgroup$ – 6005 May 13 '14 at 7:22
  • $\begingroup$ @Goos Here the equation of the circumcircle is (x^2)+(y^2)=25, so S simply represents the term '(x^2)+(y^2)-25' $\endgroup$ – HighSchoolKid May 13 '14 at 7:29

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