Imagine all the points

Geometry Level pending

A sphere of radius \(1\) is inscribed in a regular triangular pyramid \(ABCS\) with the base \(ABC.\) The angle between the base of the pyramid and its side is \(60^\circ.\) A plane intersects the edges \(AB\) and \(BC\) at some points \(M\) and \(N,\) such that \(|MN|=5,\) and it is tangent to the sphere at a point equidistant from \(M\) and \(N.\) It intersects the line of the altitude \(SK\) beyond the point \(K\) at some point \(D.\) Find \(|CD|^2.\)

Details and assumptions

A triangular pyramid \(ABCS\) is called regular if its base \(ABC\) is an equilateral triangle and the vertex \(S\) lies on the line perpendicular to the base, that passes through its center.

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