Problem A. 560. (April 2012)
A. 560. Given a right circular cone with vertex O, and a fixed point P in the base plane of the cone. Draw a line x through P which intersects the base circle of the cone at two points, X_{1} and X_{2}. Show that does not depend on the choice of the line x.
(5 pont)
Deadline expired on 10 May 2012.
Solution. Let G be the sphere with center O that contains the base circle of the cone. Let P' be the intersection of G with the ray OP, and let I be the point of G, diametrically opposite with P'.
Apply inversion to the sphere with center I and radius IP'. The image of G is the tangent plane S at point P'. The image of the base circle is some circle k in the plane S.
Let X_{1}' and X_{2}' be the images of X_{1} and X_{2}, respectively. The points P',X_{1}',X_{2}' are collinear, moreover X_{1}',X_{2}' lie on k. Then
The numerator, being the power of P' with respect to k, does not depend on the points X_{1}',X_{2}'.
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