Concurrencies and Coincidences Repost

  Steve Phelps, over at concurrencies, just wrote a "What can you do with three random points in the plane?" blog. Coincidentally, I had just finished an interesting old (1902) article about three random points on an equilateral hyperbola (such as y= 1/x for those not familiar with the term). And by another coincidence, the article happened to involve one of the common concurrent centers of a triangle, the orthocenter where the three altitudes from the vertices intersects. It turns out, that if you pick three random points on a equilateral hyperbola (they can be on either branch), then the orthocenter will also fall on the hyperbola. Stated another way, if you pick the three points all on one branch and make them all free to move, the locus of the orthocenter will be the other branch of the hyperbola.  If two points are on one branch, and one on the other is not, then the orthocenter falls on the branch with two points on it,  

Poncelet had actually written about this as far back as Jan of 1821 in Gergonne's Annales. Oh, by the way, a little "prove this factoid" for my calc kids... the y-intercept of the tangent line to any point on the rectangular hyperbola is always twice the y-coordinate, and the slope is always the square of the reciprocal of the y-coordinate. SWEET!  [Yikes, I've been busted... Keninwa noticed a mistake in the above (thanks guy) actually what I should have said (and this is only true for the basic y=1/x case), the slope is equal to the negative of the square of the y=value .. (and now, head hanging in shame, he wanders off into the sunset, muttering to himself about proofreading)..