AMC 8 2016 Problem 23

AMC8 2016 年第 23 题

圆★★★☆☆

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

两个圆心分别在点 A 和点 B 的全等圆各自通过对方的圆心。连接点 A 和点 B 的直线和这 2 个圆交于点 C 和点 D。这 2 个圆交于 2 个点，其中之一是点 E．那么的度数是多少？

正确答案 · Correct Answer

解析 · Solution

Solution 1

Observe that

\triangle{EAB}

is equilateral (all are radii of congruent circles). Therefore,

m\angle{AEB}=m\angle{EAB}=m\angle{EBA} = 60^{\circ}

. Since

CD

is a straight line, we conclude that

m\angle{EBD} = 180^{\circ}-60^{\circ}=120^{\circ}

. Since BE=BD (both are radii of the same circle),

\triangle{BED}

is isosceles, meaning that

m\angle{BED}=m\angle{BDE}=30^{\circ}

. Similarly,

m\angle{AEC}=m\angle{ACE}=30^{\circ}

. Now,

\angle{CED}=m\angle{AEC}+m\angle{AEB}+m\angle{BED} = 30^{\circ}+60^{\circ}+30^{\circ} = 120^{\circ}

. Therefore, the answer is 120.

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