AMC8 2016

AMC8 2016 · Q23

AMC8 2016 · Q23. It mainly tests Angle chasing, Circle theorems.

Two congruent circles centered at points A and B each pass through the other’s center. The line containing both A and B is extended to intersect the circles at points C and D. The two circles intersect at two points, one of which is E. What is the degree measure of $\angle CED$?

两个全等的圆分别以点A和B为中心，每个圆都经过对方的中心。包含A和B的直线延长，与圆相交于点C和D。两个圆相交于两点，其中之一是E。$\angle CED$ 的度量是多少度？

(A) 90 90

(B) 105 105

(D) 135 135

(E) 150 150

Answer

Correct choice: (C)

正确答案：（C）

Solution

We know $\triangle AEB$ is equilateral since each of its sides is a radius of one of the congruent circles. Thus the measure of $\angle AEB$ is $60^\circ$. Since $DB$ is a diameter of circle $A$ and $AC$ is a diameter of circle $B$, it follows that $\angle DEB$ and $\angle AEC$ are both right angles. Therefore the degree measure of $\angle DEC$ is $90^\circ+90^\circ-60^\circ=120^\circ$.

因为两个全等圆的半径构成了 $\triangle AEB$ 的三条边，故 $\triangle AEB$ 为等边三角形，因此 $\angle AEB=60^\circ$。又因为 $DB$ 是以 $A$ 为圆心的圆的直径，$AC$ 是以 $B$ 为圆心的圆的直径，所以 $\angle DEB$ 和 $\angle AEC$ 均为直角。于是 $\angle DEC$ 的度数为 $90^\circ+90^\circ-60^\circ=120^\circ$.

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