To construct the polar line VW of a point N with respect to a conic.

Take two secants through N: (NPR) and (NQS). Then build the complete quadrilateral (PQSRUNT). The line UT is the polar of N. The clue are the cross ratios (QSXN) and (RPWN), which are harmonic (see the file Quadrangle_0.html ). This implies that points X, Y are on the polar of N. Notice that V and W are the contact-points of the tangents to the conic from point N.

Notice that UN is the polar of point T and TN (not drawn) is the polar of U, with respect to the conic. To prove that NT is the polar of U apply the previous argument. The situation is exactly the same with the roles of N and U interchanged. Once shown that T is the intersection point of the two polars w.r. to N and U, then, by duality, follows that the polar of T passes through N and U, hence is line UN.

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