Thursday, 2 September 2010

and yet Another "Almost Pythagorean" Relation

I have written frequently about relationships that, for one reason or another, reminded me of the Pythagorean Theorem. (see here, and here, and here for example).

After trying to write up all the ones I could think of, out of the blue I got a nice note from professor Robin Whitty who maintains the excellent "Theorem of the Day" web site. (If you are a high school teacher or student, it's a must see.) He had prepared a page on Des Cartes' Circle Theorem, and had included a link to my page on the subject (I told you it was a good site). It was one more "almost Pythagorean" theorem. It's easiest to right if we consider the bend of a circle as the reciprocal of the radius, b= 1/r. With that notation adjustment, then for four circles with bends b1, b2,b3,b4, Des Cartes' Circle theorem says that one-half the square of the sum of the bends is equal to the sum of the squares of the bends.

1 comment:

Arjen Dijksman said...

Thanks for mentioning this one. It's always an excitement to discover a new one. I'll have to draw some tangent circles to understand why this works for 4 circles and not for 3 or 5.