Problem 894

Spiral of Circles

Consider a unit circle $C_{0}$ on the plane that does not enclose the origin. For $k \geq 1$ , a circle $C_{k}$ is created by scaling and rotating $C_{k - 1}$ with respect to the origin. That is, both the radius and the distance to the origin are scaled by the same factor, and the centre of rotation is the origin. The scaling factor is positive and strictly less than one. Both it and the rotation angle remain constant for each $k$ .

It is given that $C_{0}$ is externally tangent to $C_{1}$ , $C_{7}$ and $C_{8}$ , as shown in the diagram below, and no two circles overlap.

Find the total area of all the circular triangles in the diagram, i.e. the area painted green above.
Give your answer rounded to $10$ places after the decimal point.

圆形螺旋

考虑平面上一个不包含原点的单位圆 $C_{0}$ 。对于任意 $k \geq 1$ ，通过对 $C_{k - 1}$ 进行相对于原点的缩放和旋转来得到圆 $C_{k}$ 。也就是说，圆的半径和圆心到原点的距离都按相同的比例缩放，且旋转中心是原点。缩放因子为正且严格小于 $1$ 。该缩放因子和旋转角度对每个 $k$ 都保持不变。

如下图所示，已知 $C_{0}$ 与 $C_{1}$ 、 $C_{7}$ 和 $C_{8}$ 均外切，且任意两个圆不重叠。

求图中所有圆弧三角形的总面积，即上图中绿色区域的总面积。
将你的答案四舍五入保留 $10$ 位小数。