Let P be the point (a, b) with 0 < b < a. Find Q on the x-axis and R on y = x, so that PQ + QR + RP is as small as possible.
Solution
Easy.
Reflect P in the x-axis to get A (a, -b) and in the line y = x to get B (b, a). Then PQ = AQ and RP = RB, so PQ + QR + RP = AQ + QR + RB ≥ AB, with equality iff Q and R lie on AB. So the minimum is √(2a2 + 2b2).
© John Scholes
jscholes@kalva.demon.co.uk
12 Dec 1998