Poncelet's closure theorem and three levels of its understanding

Poncelet's 1814 theorem 

states that whenever a triangle is inscribed in one circle and circumscribed another one, the triangle belongs to the continuous

family of triangles that are all inscribed in and circumscribe the same two circles.

This theorem will be presented as

(1) an "obvious" fact of elementary geometry;

(2) as a rather difficult high-school theorem;

(3) as a quite natural result of adult mathematics.

The level (3) needs some understanding of complex algebraic geometry. However, no preliminary knowledge of it is assumed and all

the necessary concepts will be explained.