IMO 1977 LL USA52

Origin: USA

Problem

Two perpendicular chords are drawn through a given interior point P of a circle with radius R. Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from P to the center of the circle is kR.

Solution

The maximum and minimum are 2R \sqrt 4 −2k2 and 2R
1 + \sqrt 1 −k2 re- spectively.