2017 ACM-ICPC 亚洲区（南宁赛区）网络赛： G. Finding the Radius for an Inserted Circle（笛卡尔定理）

Three circles C​a​​, C​b​​, and C​c​​, all with radius R and tangent to each other, are located in two-dimensional space as shown in Figure 1. A smaller circle C​1​​ with radius R​1​​ (R​1​​<R) is then inserted into the blank area bounded by C​a​​, C​b​​, and C​c​​ so that C​1​​ is tangent to the three outer circles, C​a​​, C​b​​, and C​c​​. Now, we keep inserting a number of smaller and smaller circles C​k​​ (2≤k≤N) with the corresponding radius R​k​​ into the blank area bounded by C​a​​, C​c​​ and C​k−1​​ $(2\le k\le N)$, so that every time when the insertion occurs, the inserted circle C​k​​ is always tangent to the three outer circles C​a​​, C​c​​ and C​k−1​​, as shown in Figure 1

Figure 1.

(Left) Inserting a smaller circle C​1​​ into a blank area bounded by the circle C​a​​, C​b​​ and C​c​​.

(Right) An enlarged view of inserting a smaller and smaller circle C​k​​ into a blank area bounded by C​a​​, C​c​​ and C​k−1​​ ($2\le k\le N$), so that the inserted circle C​k​​ is always tangent to the three outer circles, C​a​​, C​c​​, and C​k−1​​.

Now, given the parameters R and k, please write a program to calculate the value of R​k​​, i.e., the radius of the k-th inserted circle. Please note that since the value of R​k​​ may not be an integer, you only need to report the integer part of R​k​​. For example, if you find that R​k​​ = $1259.8998$ for some k, then the answer you should report is $1259$.

Another example, if R​k​​ = $39.1029$ for some k, then the answer you should report is $39$.

Assume that the total number of the inserted circles is no more than 10, i.e., $N\le 10$. Furthermore, you may assume $\pi =3.14159$. The range of each parameter is as below:

$1\le k\le N$, and 10​4​​≤R≤10​7​​.

Input Format

Contains l + 3 lines.

Line 1: l ----------------- the number of test cases, l is an integer.

Line 2: $R$ ---------------- R is a an integer followed by a decimal point,then followed by a digit.

Line 3: k ---------------- test case #1, k is an integer.

$\dots$

Line i+2: k ----------------- test case # i.

$\dots$

Line l +2: k ------------ test case #l.

Line l + 3: -1 ---------- a constant -1 representing the end of the input file.

Output Format

Contains $l$lines.

Line 1: k R​k​​ ----------------output for the value of k and R​k​​ at the test case #1, each of which should be separated by a blank.

$\dots$

Line i: k R​k​​ ----------------output for k and the value of R​k​​ at the test case # i, each of which should be separated by a blank.

Line l: k R​k​​ ----------------output for k and the value ofR​k​​ at the test case # l, each of which should be separated by a blank.

样例输入

1
152973.6
1
-1

样例输出

1 23665

思路：

(1)笛卡尔定理

定义一个圆的曲率k=+1/r或-1/r，其中r是其半径。 

若平面有两两相切，且有6个独立切点的四个圆，设其曲率为k1,k2,k3,k4（若该圆与其他圆均外切，则曲率取正，否则取负）则其满足性质： 

                                         （k1+k2+k3+k4）*（k1+k2+k3+k4）=2*（k1*k1+k2*k2+k3*k3+k4*k4）

根据这个定理，我们就可以类似迭代的方式，不断往后递推求解。

#include
using namespace std;
const double PI=3.14159;
int main()
{
    int T,n;
    double R;
    while(scanf("%d",&T)!=EOF&&T!=-1)
    {
        scanf("%lf",&R);
        while(T--)
        {
            scanf("%d",&n);
            double k1=1/R,k2=1/R,k3=1/R;
            double ans;
            for(int i=0;i