Question

在附图中，五个圆依次彼此相切，并且都与直线 $L_1$ 和 $L_2$ 相切。若最大圆的半径为 $18$ ，最小圆的半径为 $8$ ，则中间圆的半径为

In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2.$ If the radius of the largest circle is $18$ and that of the smallest one is $8,$ then the radius of the middle circle is

$12$

$12.5$

$13$

$13.5$

$14$

答案：A

知识点：相切圆相似等比数列

难度评级：2310

小提示：

任意两个相邻相切圆的图形缩放后形状相同

Every pair of consecutive tangent circles has the same shape after scaling

大提示：

这些半径组成等比数列，所以中间半径的平方等于两个极端半径的乘积

The radii form a geometric sequence, so the middle radius squared is the product of the extremes

解答：

各圆心都位于 $L_1$ 与 $L_2$ 的角平分线上。任意两个相邻圆的构型都相似，因此相邻半径之比为常数。所以五个半径组成等比数列。若中间半径为 $m$ ，五项等比数列的对称性给出 $m^2=8\cdot18$ ，所以 $m=12$ 。

所以正确答案是 A。

The centers lie on the angle bisector of $L_1$ and $L_2.$ Similarity of the configuration for any two consecutive circles shows that consecutive radii have a constant ratio. Thus the five radii form a geometric sequence. If the middle radius is $m,$ symmetry of a five-term geometric sequence gives $m^2=8\cdot18,$ so $m=12.$

Therefore, the correct answer is A.

1983 AMC 12 第 23 题

答案：A

难度评级：2310

小提示：

大提示：

解答：

其他年份的第 23 题