Triangle, circumcircle, perpendicular bisectors...
Source: DeMO 2005, classes 11 and 12/13, 1st day, problem 2
May 9, 2005
geometrycircumcircleperpendicular bisectorgeometry proposed
Problem Statement
According to the estimated number of participants who gave a correct solution, this was the hardest (!) problem from today's paper. So here is this great German killer - be warned!
Given a circle k and three pairwisely distinct points A, B, C on this circle. Let h and g be the perpendiculars to the line BC at the points B and C. The perpendicular bisector of the segment AB meets the line h at a point F; the perpendicular bisector of the segment AC meets the line g at a point G.
Prove that the product is independent from the position of the point A, as long as the points B and C stay fixed.
The actual problem behind the problem: Why on hell should the points B and C stay fixed?
Darij