ABC is an equilateral triangle. Let PQRS be a square inscribed in it such that P is on AB and Q is on AC. Which of the following is/are correct? I. AP : PB = 4 : 3 II. √3 AB = (2 + √3) PQ

Let the side of equilateral triangle = a and side of square = s. The square PQRS has PQ on... wait, with P on AB and Q on AC, and the square inscribed with R, S on BC. By geometry of equilateral triangle, side of square s = a·√3/(2 + √3) = a√3(2 − √3) = a(2√3 − 3). Then √3·a = √3·a, and (2 + √3)s = (2 + √3)·a√3/(2 + √3) = a√3 = √3·AB. So II is correct. For I: AP corresponds to position where the inscribed square meets AB. By geometry, AP = (a − s)/2·... actually AP:PB is symmetric due to symmetry of equilateral triangle, giving AP = PB, ratio 1:1, not 4:3. So I is incorrect. Only II is correct.