HMMT 二月 2001 · 几何 · 第 6 题

英文解析

6. A point on a circle inscribed in a square is 1 and 2 units from the two closest sides ofthe square. Find the area of the square.
   Solution: Call the point in question A , the center of the circle O , and its radius r .
   Consider a right triangle BOA with hypotenuse OA : OA has length r , and BO and BA
   2 2 2
   have lengths r − 1 and r − 2. By the Pythagorean theorem, ( r − 1) + ( r − 2) = r ⇒
   2 2
   r − 6 r + 5 = 0 ⇒ r = 5 since r > 4. The area of the square is (2 r ) = 100 .