Some results on quadrilaterals 
The Inscribed Quadrilateral Theorem
Proof of the Inscribed Quadrilateral Theorem This suggests a construction for the polar of a point P: 
The dual of this is
The Circumscribed Quadrilateral Theorem
Here, we may observe that to say that a point P is the pole of a line QR
If a quadrilateral is inscribed in, or circumscribed round a conic,

Indeed, corresponding inscribed and circumscribed quadrilaterals
The Dual Quadrilaterals Theorem
Proof of the Dual Quadrilaterals Theorem

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