Drawing a Character Face

Time limit0.1sMemory limit256 MB

Summary

Given three circles, compute the area of their union, counting overlaps once, and print it to six decimal places.

Level: Hard8 of 10
Topics: Geometry, Math, Implementation, Brute force
Solved: No attempts yet

Problem

Many character faces can be drawn with three circles. When the face is one circle and the two ears sticking out at the sides are circles as well, laying three circles on top of each other already gives the rough outline of the face. The mouse character, the lion character, the tiger character and the bear character all work this way.

Silhouette of a character face drawn with three circles

You want the area of the figure drawn this way. Given the center and the radius of each of the three circles, write a program that computes the area of the region the three circles cover. Area where two circles overlap counts once.

Input

The input has three lines. Each line holds the integers $x$ , $y$ and $r$ , separated by spaces. They are the $x$ coordinate of the center, the $y$ coordinate of the center and the radius of the circle, in that order.

Every circle lies inside the square whose opposite corners are $(0, 0)$ and $(100, 100)$ , so $r \le x \le 100 - r$ and $r \le y \le 100 - r$ . The radius is at least $1$ .

Two of the three circles can sit at the same place with the same radius, and one circle can lie completely inside another.

Output

Print the area of the region the three circles cover on one line, rounded to six digits after the decimal point. Always print all six digits.

Examples2

Example 1
Input
```
10 10 5
5 15 3
15 15 3
```
Expected output
```
130.570546
```
Example 2
Input
```
30 40 23
70 40 23
50 74 23
```
Expected output
```
4683.733435
```