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# The inner parallel

A cyclic quadrilateral with its diagonals and two altitudes. Prove that AB is parallel to EF.

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## 5 replies on “The inner parallel”

Marshall Bucksays:

The following Lemma (in the diagram) can be used for a proof of this problem.
The line connecting the bases of the altitudes is parallel to the top edge of the quadrilateral, so the Lemma applies.

Marshall Bucksays:

I do not yet have a proof of the Lemma, but it appears to be a result in Projective Geometry and generalizes as shown in this diagram.

Marshall Bucksays:

With projective geometry, it suffices to prove the Lemma in the case of 3 parallel lines intersecting 3 other parallels, perpendicular to the first set. Then, the result can be shown using simple coordinate algebra.

Marshall Bucksays:

This is just the projective dual of the famous Pappus’s Theorem, so nothing new at all. Oh, well…

It looked familiar! Indeed, Pappus strikes again.