1983 AMC 12 第 23 题

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：

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