HMMT 二月 2002 · 冲刺赛 · 第 32 题

英文解析

32. Two circles have radii 13 and 30, and their centers are 41 units apart. The linethrough the centers of the two circles intersects the smaller circle at two points; let A be theone outside the larger circle. Suppose B is a point on the smaller circle and C a point onthe larger circle such that B is the midpoint of AC . Compute the distance AC .
    √
    Solution: 12 13 Call the large circle’s center O . Scale the small circle by a factor of
    2 about A ; we obtain a new circle whose center O is at a distance of 41 − 13 = 28 from O ,1
    2 1
    and whose radius is 26. Also, the dilation sends B to C , which thus lies on circles O and O .
    1 2
    So points O , O , C form a 26-28-30 triangle. Let H be the foot of the altitude from C to
    1 2
    √ √
    2 2
    O O ; we have CH = 24 and HO = 10. Thus, HA = 36, and AC = 24 + 36 = 12 13.
    1 2 2