On the Relation Between a Locus and Poncelet’s Closure Theorem

This article contains a synthetic proof of the following proposition: consider a conic c1 and its variable chord AB, which subtends a right angle at a given point P. Then, the foot E of the perpendicular dropped from P onto the line AB lies on a certain circle (the line being the limiting case of the circle). To prove this proposition, we show how Poncelet’s closure theorem for quadrilaterals can be derived by elementary projective considerations only (without any computations, either in Cartesian or projective coordinates). Finally, the limiting case of the proposition, where the point P lies on the conic, is also mentioned. The problem can then be reduced to Frégier’s theorem and may represent a slightly different perspective on this theorem.