On 9/24/06, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
Take any triangle, and trisect the three edges. Connect each corner to the left trisection mark. What is the ratio of the full triangle to the interior triangle?
The name Harold McIntosh forgets is, according to De Villiers, M. (2005). Feedback: Feynman's Triangle. The Mathematical Gazette, 89 (514), March 2005, p. 107; see also the more extended version at http://mysite.mweb.co.za/residents/profmd/homepage4.html where he proves the extension to 1/p-section of each edge, the area of the inner triangle now being (p-2)^2/(p^2-p+1) of the outer.
Draw 3 circles that are tangent to each other, and which are also tangent to a line. Draw a unit circle tangent to the three other circles. What is the distance from the center of the unit circle to the line?
Multiple tangencies have to be excluded, as Jason Holt (somewhat cryptically) points out. The result is easy to prove by inversion in the circle centred at the point where the third circle meets the line, radius chosen so orthogonal to the unit radius circle. The result comprises two tangent circles of radius 4, two lines tangent to both of them, and the unaltered unit circle with centre obviously distant 7 from the unaltered line.
What are some of the other amazing appearances of the number 7?
Ed Pegg Jr
See The Seven Circles Theorem by Stanley Rabinowitz, Pi Mu Epsilon Journal, 8(1987)441–449 (available on the web) where he proves "Start with a circle. Any circle. Draw six more circles inside it, each internally tangent to the original circle and tangent to each other in pairs. Let A, B, C, D, E, and F be the consecutive points of tangency of the small circles with the outer circle. We wind up with a set of seven circles as shown in Figure 1. The Seven Circles Theorem says that no matter what sizes we pick for the seven circles (subject only to certain order and tangency constraints), it will turn out that the lines AD, BE, and CF will meet in a point." Fred Lunnon