如图所示,在平面四边形\(ABCD\)中,\(AB=1\),\(BC=2\),\(\triangle ACD\)为正三角形,则\(\triangle BCD\)面积的最大值为\(\underline{\qquad\qquad}\).
![](https://img2018.cnblogs.com/blog/1793042/201910/1793042-20191009201017141-1835227188.png)
解析: 将\(BC\)边固定,则\(A\)点在以\(B\)为圆心,\(1\)为半径的圆上运动, 由于\(\triangle ACD\)为正三角形,因此
![](https://img2018.cnblogs.com/blog/1793042/201910/1793042-20191009201023946-872722903.png)
\(D\)点是在以\(E\)点为圆心,\(1\)为半径的圆上运动,其中\(E\)点是把\(B\)点绕着\(C\)点逆时针旋转\(60^\circ\)所得的点.因此显然当\(D\)点位于圆\(E\)的上端顶点时,\(\triangle BCD\)的面积最大,且此时面积最大值为$$
S=\dfrac{1}{2}\cdot |BC|\cdot \left(\dfrac{\sqrt{3}}{2}|BC|+1\right)=\sqrt{3}+1.$$