If a hexagon ABCDEF is circumscribed around a conic and the midpoints P, Q and R of the alternate sides AB, CD & EF touch the conic, and the other three points where the remaining sides of the hexagon touches the conic are labelled X, Y and Z as shown, then the lines PY, XR & QZ connecting opposite tangential points are concurrent at the Brianchon point.
Corollary 1: For the hexagon above, sin A * sin C * sin E = sin B * sin D * sin F.
Corollary 2: The hexagon ABCDEF is inscribed in a conic.
(Drag any of the 'bright' red points).
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Download the dynamic geometry software Cinderella 2 for FREE from here, and use it (after unzipping) to view & manipulate the Cinderella BMO conic general dynamic sketch illustrating the above result.
Challenge: Can you explain why (prove) the result is true? If stuck, have a look at the 2005 paper of my friend & colleague, Michael Fox, from the United Kingdom at Proof of BMO conic general which provides not only a proof, but a further generalization. A dynamic sketch by Michael Fox that illustrates the proof and generalization is at Michael Fox theorem.
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Created by Michael de Villiers, 30 Oct 2010 with Cinderella; updated 15 October 2021; 7 April 2023.