Talk:Five circles theorem

Unclear

This article currently says:

In geometry, the five circles theorem states that, given five circles centered on a common sixth circle and intersecting each other chainwise on the same circle, the lines joining the their second intersection points forms a pentagram whose points lie on the circles themselves.

What does "centered on" mean? I'm accustomed to "centered at (a point)", but "centered on (a circle)" I don't understand. Here's wild guess: "centered on a common sixth circle" might mean externally tangent to a common sixth circle.

What does "second intersection points" mean? There are intersection points where they touch each other chainwise and there may be points of tangency to the common sixth circle; does "second intersection points" refer to one of those?

What does "whose points lie on circles" mean? My guess is it means "whose vertices lie on a circle", but I'm not at all sure of that.

I assume "chainwise" means in the sense of Steiner chains or Pappus chains. Should we link to one of those, or is there some other appropriate article to link to?

Michael Hardy (talk) 13:49, 5 October 2009 (UTC)

On the one hand, most of these questions are addressed by the article at Wolfram Mathworld. On the other hand, the statement of the theorem given at theoremoftheday.org seems to be the converse of the statement here. The French Wikipedia has a short article about Miquel with links to his publications, but I don't have time right now to search for the original statement of the theorem. Jowa fan (talk) 05:31, 5 August 2011 (UTC)