The real numbers x1, x4, y1, y2 are positive and the real numbers x2, x3, y3, y4 are negative. We have (xi - a)2 + (yi - b)2 ≤ c2 for i = 1, 2, 3, 4. Show that a2 + b2 ≤ c2. State the result in geometric language.
Solution
Stated geometrically, the result is: if a disk includes a point in each quadrant, then it must also include the origin. We use the fact that a disk is convex. Let Pi be the point (xi,yi). The segment P1P2 must intersect the positive x-axis. By convexity, the point of intersection, call it X, must lie in the disk. Similarly, P3P4 must intersect the negative x-axis at some point Y, which must be in the disk. Then all points of the segment XY are in the disk and hence, in particular, the origin.
© John Scholes
jscholes@kalva.demon.co.uk
7 March 2004
Last corrected/updated 7 Mar 04