Problem 1533 (difficulty: 9/10)

Consider \(\displaystyle \C\) as the \(\displaystyle xy\)-plane in 3-space and pick 2 semicircles in the upper half space whose end points are the complex numbers \(\displaystyle a,b\) and \(\displaystyle c,d\). Show that the two semicircles intersect each other orthogonally if and only if \(\displaystyle (a,b,c,d)=-1\).

(Riesz competition, 1988)

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