Second solution: For any λ > 0, the map (x,y) 7→ (λx,λ−1y) preserves both areas and the hyperbola xy = 1. We may thus rescale the picture so that A,B are symmetric across the line y = x, with A above the line. As P moves from A to B, the area of APB increases until P passes through the point (1,1), then decreases. Consequently, P = (1,1) achieves the maximum area, and the desired equality is ob...