# 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.

×