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, X1 and X2. Show that does not depend on the choice of the line x.
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 X1' and X2' be the images of X1 and X2, respectively. The points P',X1',X2' are collinear, moreover X1',X2' lie on k. Then
The numerator, being the power of P' with respect to k, does not depend on the points X1',X2'.