14 Dec
Circles, Products and Optimization
Here's a lively problem that was offered at the 1975 USA Math Olympiad (Porblem 4): Circle C(E), with center E, and C(F) with center F, meet in points P and Q. A on C(E) and B on C(F) are such that AB passes through P. Find the position of A and B for which AP*PB is maximum. The curious thing is that there are a trigonometric solution and a synthetic solution that give two different constructions, but I do not see any relation between the two.
The problem led to several discussions:
- Problem 4, 1975 USA Math Olympiad and Isosceles Triangles
- Problem 4, 1975 USA Math Olympiad: Normals and Tangents
- Two Circles and One More
- Problem 4, 1975 USA Math Olympiad and the Radical Axis
No related posts.
Found it: http://www.cut-the-knot.org/Curriculum/Geometry/MaxCommonChord6.shtml
December 17th, 2009 at 1:32 pm