On a conjecture of Kontsevich and variants of Castelnuovo's lemma

Let A = (a(j)(i)) be an orthogonal matrix (over R or C) with no entries zer o. Let B = (b(j)(i)) be the matrix defined by b(j)(i) = 1/a(j)(i). M. Konts evich conjectured that the rank of B is never equal to three. We interpret this conjecture geometridally and prove it. The geometric statement can be understood as variants of the Castelnuovo lemma and Brianchon's theorem.