2016 AMC 8 第 23 题

Question

两个全等圆分别以点

A

和

B

为圆心，并且每个圆都经过另一个圆的圆心。经过

A

和

B

的直线延长后与两个圆分别交于点

C

和

D

。

两圆相交于两点，其中一点为

E

。

\angle CED

的度数是多少？

Two congruent circles centered at points

A

and

B

each pass through the other circle’s center. The line containing both

A

and

B

is extended to intersect the circles at points

C

and

D.

The circles intersect at two points, one of which is

E.

What is the degree measure of

\angle CED?

90

105

120

135

150

答案：C

知识点：等边三角形圆周角导角

难度评级：1510

小提示：

两个圆心和一个交点构成等边三角形

The centers and one intersection point form an equilateral triangle.

大提示：

使用直径所对圆周角是直角这一事实

Use the fact that angles subtending diameters are right angles.

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文字解答：

因为

AE = EB = AB

，它们都是全等圆的半径，所以三点构成等边三角形，

\angle AEB = 60^{\circ}

。

又因为

\overline{DB}

和

\overline{AC}

是直径，所以

\angle DEB=\angle AEC=90^\circ

。因此

\begin{aligned} \angle CED &= \angle DEB+\angle AEC\\ &\qquad-\angle AEB\\ &=90^\circ+90^\circ-60^\circ\\ &=120^\circ\text{。} \end{aligned}

所以正确答案是 C。

We know that

AE = EB = AB

since they are all radii of congruent circles, so they form an equilateral triangle, which means that

\angle AEB = 60^{\circ}.

The segments

\overline{DB}

and

\overline{AC}

are diameters, so

\angle DEB=\angle AEC=90^\circ.

Therefore,

\begin{aligned} \angle CED &= \angle DEB+\angle AEC\\ &\qquad-\angle AEB\\ &=90^\circ+90^\circ-60^\circ\\ &=120^\circ. \end{aligned}

Thus, C is the correct answer.