4th Eötvös 1897 Problem 3 The line L contains the distinct points A, B, C, D in that order. Construct a rectangle whose sides (or their extensions) intersect L at A, B, C, D and such that the side which intersects L at C has length k. How many such rectangles are there?
Draw a line through B parallel to the rectangle sides length k. Suppose it meets the side through A at X. Then ∠AXB = 90o and BX = k. So if AB ≤ k then there is no rectangle. If AB > k, then X must lie on the circle diameter AB and on the circle center B radius k. There are two such points and hence two possible rectangles. Having constructed X, the rest follows easily. 4th Eötvös 1897 © John Scholes jscholes@kalva.demon.co.uk 29 Oct 2003 Last corrected/updated 29 Oct 03